AU774975B2 - Method and apparatus for determining the area enclosed by a planar curve - Google Patents

Description

S&FRef: 564839

AUSTRALIA

PATENTS ACT 1990 COMPLETE SPECIFICATION FOR A STANDARD PATENT

ORIGINAL

Name and Address of Applicant Actual Inventor(s): Address for Service: Invention Title: Canon Kabushiki Kaisha 30-2, Shimomaruko 3-chome, Ohta-ku Tokyo 14 6 Japan Cameron Bolitho Browne Spruson Ferguson St Martins Tower,Level 31 Market Street Sydney NSW 2000 (CCN 3710000177) Method and Apparatus for Determining the Area Enclosed by a Planar Curve ASSOCIATED PROVISIONAL APPLICATION DETAILS [33] Country [31] Applic. No(s) AU PQ9389 [32] Application Date 14 Aug 2000 The following statement is a full description of this invention, including the best method of performing it known to me/us:- 5815c -1- METHOD AND APPARATUS FOR DETERMINING THE AREA ENCLOSED BY A PLANAR CURVE Technical Field of the Invention The present invention relates generally to the field of computational geometry and, in particular, to determining the area enclosed by a planar curve. The present invention relates to a method and apparatus for determining the area enclosed by a planar curve. The invention also relates to a computer program product including a computer readable medium having recorded thereon a computer program for determining the area enclosed by a planar curve.

Background In the field of computational geometry it is often required to determine the area enclosed by a planar curve. For example, when performing the task of packing a region defined by an outline boundary with circles of a specified size, it may be required to perform an estimate of how many circles need to be packed in total in order to estimate the progress of the operation at regular intervals. Estimating how many circles will be packed into the boundary region in total requires that the area enclosed by the boundary region be known.

There are several well known computational methods for calculating the area enclosed by a polygon or by a Bezier curve. For example, the area of a simple polygon can be found by the formula: A= X/ x2 xn XI *y Y2 Yn y where (xi, yi), i n represents the counter clockwise enumeration of the vertices of "the polygon, and where the interpretation of x 2 x x, Yl Y2 Yn Yl is the summing of the products of the "downwards" diagonals and subtraction of the product of the "upwards" diagonals. Further, a first estimate of the area enclosed by a collection of Bezier or Bspline curves can be determined using the integral: Y P(t)x P'(t)dt 564839.doc 28. MAY. 2004 13:36 SPRUSON IFERGUSON NO. 5916 P. 4 -2where P(t) represents the curve as a function of time.

However, existing computational methods for area calculation based on the above-formulas and on other known methods, tend to be either fast and inaccurate or precise but slow.

Disclosure of the Invention It is an object of the present invention to substantially overcome, or at least iameliorate, one or mone disadvantages of existing arrangements, According to one aspect of the present invention there is provided a method of displaying a planar curve image based on the area enclosed by a planar curve defining the image, said mnethod compoising the steps of dividing said curve into at least one curve segment; determining an area enclosed by a polygon based on said at least one curve segment; *~*determining a remaining area enclosed by each of said curve segments, said remaining area being determined by iteratively sampling each of said curve segments to a predetennined interval to provide at least one sample region for each of said curve segments, and determining an area defined by each of said sample regions and a reference line associated with each of said curve segments; summing said area enclosed by said polygon and said remaining area to deterinine the area enclosed by said planar curve; and displaying the planar curve image in accordance with the determined area enclosed by said planar curve.

According to another aspect of the present invention there is provided a method displaying a planar curve image based on the area enclosed by a planar curve defining 25 the image, said method comprising the steps of: dividing said curve into at least one curve segment; deternining a polygon based on said at least one curve segment; determining an area enclosed by said polygon; determining a remaining area enclosed by each of said curve segments, said determining comprising for each said curve segment, the sub-steps of: (Da) sampling said curve segment at a predetermined interval to provide at least one sample region; (Db) determining an area defined by each sample region and a reference line associated with each of said curve segments; 56483Q.doe COMs ID No: SBMI-00769912 Received by IP Australia: Time 13:46 Date 2004-05-28 28. MAY. 2064 13:36 SPRUSON FERGUSON NO.5916 P. (Dc) repeating steps (Da) and (Db) a predetermined number of times;, (Dd) summing said areas defined by each sample region to provide said curve segment area; summing said area enclosed by said polygon and each of said curve segment areas to determine the area enclosed by said planar curve; and displaying the planar curve image in accordance with the determined area enclosed by said planar curve.

According to another aspect of the present invention there is provided a method of displaying a planar curve image based on the area enclosed by a planar curve defining the image, said method comprising the steps of: dividing said curve into at least one Bezier curve segment; determining a polygon based on said at least one Bezier curve segment; determining an area enclosed by said polygon; detennining a remaining area enclosed by each of said Bezier curve 159 segments, said determining comprising for each said Bezier curve segiment, the sub-steps: (Da) sampling said curve segment at a predetermined interval to provide at least one sample region; (Db) determining an area defined by each sample region and an associated reference line; (Dc) repeating steps (Da) and (Db) a predetermined number of times; (Dd) summing said areas defined by each sample region to provide said curve segment area; summing said area enclosed by said polygon and each of said Bezier curve 9. 9 segment areas to determine the area enclosed by said planar curve displaying the planar curve image in accordance with the determined area enclosed by said planar curve.

According to still another aspect of the present invention there is provided an apparatus for determining the area enclosed by a planar curve, said apparatus comprising: dividing means for dividing said curve into at least one curve segment; determining means for determining an area enclosed by a polygon based on said at least one curve segment, and for determining a remaining area enclosed by each of said curve segments, said remaining area being determined by iteratively sampling each of 564839.doc COMS ID No: SBMI-00769912 Received by IP Australia: Time 13:46 Date 2004-05-28 said curve segments to a predetermined interval to provide at least one sample region for each of said curve segments, and determining an area defined by each of said sample regions and a reference line associated with each of said curve segments; and summing means for summing said area enclosed by said polygon and said remaining area to provide the area enclosed by said planar curve.

According to still another aspect of the present invention there is provided an apparatus for determining the area enclosed by a planar curve, said apparatus comprising: means for dividing said curve into at least one curve segment; means for determining a polygon based on said at least one curve segment; means for determining an area enclosed by said polygon; means for determining a remaining area enclosed by each of said curve segments, said means for determining further comprising: means for sampling each said curve segment at a predetermined interval to provide at least one sample region; 15 (ii) means for determining an area defined by each sample region and a reference line associated with each of said curve segments; o (iii) means for summing said areas defined by each sample region to provide said curve segment area; and means for summing said area enclosed by said polygon and each of said curve segment areas to provide the area enclosed by said planar curve.

According to still another aspect of the present invention there is provided an apparatus for determining the area enclosed by a planar curve, said apparatus comprising: S-a memory for storing a program; 25 a processor for executing the program, said program being configured to perform the following steps: dividing said curve into at least one curve segment; determining an area enclosed by a polygon based on said at least one curve segment; 564839.doc determining a remaining area enclosed by each of said curve segments, said remaining area being determined by iteratively sampling each of said curve segments to a predetermined interval to provide at least one sample region for each of said curve segments, and determining an area defined by each of said sample regions and a reference line associated with each of said curve segments; and summing said area enclosed by said polygon and said remaining area to provide the area enclosed by said planar curve.

According to still another aspect of the present invention there is provided an apparatus for determining the area enclosed by a planar curve, said apparatus comprising: a memory for storing a program; a processor for executing the program, said program being configured to perform the following steps: dividing said curve into at least one curve segment; determining a polygon based on said at least one curve segment; determining an area enclosed by said polygon; determining a remaining area enclosed by each of said curve segments, said determining comprising for each said curve segment, the sub-steps of: (Da) sampling said curve segment at a predetermined interval to provide at least one sample region; (Db) determining an area defined by each sample region and a reference line associated with each of said curve segments; (Dc) repeating steps (Da) and (Db) a predetermined number of times; (Dd) summing said areas defined by each sample region to provide said curve segment area; and 564839.doc summing said area enclosed by said polygon and each of said curve segment areas to provide the area enclosed by said planar curve.

According to still another aspect of the present invention there is provided an apparatus for determining the area enclosed by a planar curve, said apparatus comprising: a memory for storing a program; a processor for executing the program, said program being configured to perform the following steps: dividing said curve into at least one Bezier curve segment; determining a polygon based on said at least one Bezier curve segment; determining an area enclosed by said polygon; determining a remaining area enclosed by each of said Bezier curve segments, said determining comprising for each said Bezier curve segment, the substeps: (Da) sampling said curve segment at a predetermined S"interval to provide at least one sample region; (Db) determining an area defined by each sample region and an associated reference line; :e i (Dc) repeating steps (Da) and (Db) a predetermined number of times; S" summing said areas defined by each sample region to go 4 provide said curve segment area; and summing said area enclosed by said polygon and each of said Bezier curve segment areas to provide the area enclosed by said planar curve.

According to still another aspect of the present invention there is provided a program for determining the area enclosed by a planar curve, said program comprising: code for dividing said curve into at least one curve segment; 564839.doc code for determining an area enclosed by a polygon based on said at least one curve segment; code for determining a remaining area enclosed by each of said curve segments, said remaining area being determined by iteratively sampling each of said curve segments to a predetermined interval to provide at least one sample region for each of said curve segments, and determining an area defined by each of said sample regions and a reference line associated with each of said curve segments; and code for summing said area enclosed by said polygon and said remaining area to provide the area enclosed by said planar curve.

According to still another aspect of the present invention there is provided a program for determining the area enclosed by a planar curve, said program comprising code for performing the following steps: dividing said curve into at least one curve segment; determining a polygon based on said at least one curve segment; 15 determining an area enclosed by said polygon; determining a remaining area enclosed by each of said curve segments, said S"determining comprising for each said curve segment, the sub-steps of: o (Da) sampling said curve segment at a predetermined interval to provide at least one sample region; S° 20 (Db) determining an area defined by each sample region and a reference line associated with each of said curve segments; repeating steps (Da) and (Db) a predetermined number of times; (Dd) summing said areas defined by each sample region to provide said curve segment area; and summing said area enclosed by said polygon and each of said curve segment areas to provide the area enclosed by said planar curve.

According to still another aspect of the present invention there is provided a program for determining the area enclosed by a planar curve, said program comprising code for performing the following steps: 564839.doc dividing said curve into at least one Bezier curve segment; determining a polygon based on said at least one Bezier curve segment; determining an area enclosed by said polygon; determining a remaining area enclosed by each of said Bezier curve segments, said determining comprising for each said Bezier curve segment, the sub-steps: (Da) sampling said curve segment at a predetermined interval to provide at least one sample region; (Db) determining an area defined by each sample region and an associated reference line; (Dc) repeating steps (Da) and (Db) a predetermined number of times; (Dd) summing said areas defined by each sample region to provide said curve segment area; and summing said area enclosed by said polygon and each of said Bezier curve segment areas to provide the area enclosed by said planar curve.

Brief Description of the Drawings One or more embodiments of the present invention will now be described with reference to the drawings, in which: Fig. 1 shows the area enclosed by a planar curve; 20 Fig. 2 shows the curve of Fig. 1 described as a number of cubic Bezier segments with corresponding control polygons; Fig. 3 shows a curve described by a single cubic Bezier segment; Fig. 4 shows an area enclosed by a polygon formed by the on-curve points of the closed planar curve of Fig. 1; 25 Fig. 5 shows the area enclosed by the projection of a line segment (AB) of the polygon of Fig. 4; Fig. 6 shows the remaining area enclosed by the curve of Fig. 1 after the polygonal area of Fig. 4 has been determined in accordance with one arrangement; Fig. 7 shows a curve segment and a reference line; 564839.doc Fig. 8 shows the curve segment of Fig. 7 sub-divided into two sub-curves; Fig. 9 shows the curve segment of Fig. 7 sampled at regular intervals; Fig. 10 shows the curve segment of Fig. 7 sampled at regular intervals; Fig. 11 shows the curve segment of Fig. 7 described along interval t=O to t=l; Fig. 12 shows the curve segment of Fig. 7 sampled at the points t=0.25, t=0.75, and t=l.0; Fig. 13 shows the curve segment of Fig. 7 sampled at the points t=0.125, t=0.375, t=0.625, and t=0.875; Fig 14 shows an open planar curve described by a piecewise cubic Bezier segment and corresponding control polygon, in accordance with a further arrangement; Fig. 15 shows a first approximation to the area enclosed by the curve of Fig. 14; Fig. 16 is a flowchart showing a method of determining the area enclosed by a planar curve, in accordance with one arrangement; Fig. 17 is a flowchart showing a method of determining the area enclosed by a curve segment and a reference line; and Fig. 18 is a schematic block diagram of a general purpose computer upon which S•the embodiments of the present invention can be practiced.

oooo Detailed Description including Best Mode Where reference is made in any one or more of the accompanying drawings to .ooo.i 20 steps and/or features, which have the same reference numerals, those steps and/or features have for the purposes of this description the same function(s) or operation(s), unless the S" "contrary intention appears.

Some portions of the description which follows are explicitly or implicitly presented in terms of algorithms and symbolic representations of operations on data .25 within a computer memory. These algorithmic descriptions and representations are the means used by those skilled in the data processing arts to most effectively convey the substance of their work to others skilled in the art. An algorithm is here, and generally, conceived to be a self-consistent sequence of steps leading to a desired result. The steps are those requiring physical manipulations of physical quantities. Usually, though not 564839.doc necessarily, these quantities take the form of electrical or magnetic signals capable of being stored, transferred, combined, compared, and otherwise manipulated. It has proven convenient at times, principally for reasons of common usage, to refer to these signals as bits, values, elements, symbols, characters, terms, numbers, or the like.

It should be borne in mind, however, that the above and similar terms are to be associated with the appropriate physical quantities and are merely convenient labels applied to these quantities. Unless specifically stated otherwise, and as apparent from the following, it will be appreciated that throughout the present specification, discussions utilizing terms such as "scanning", "calculating", "determining", "replacing", "generating" "initializing", "outputting", or the like, refer to the action and processes of a computer system, or similar electronic computing device, that manipulates and transforms data represented as physical (electronic) quantities within the registers and memories of the computer system into other data similarly represented as physical quantities within the computer system memories or registers or other such information storage, transmission or display devices.

The present specification also discloses apparatus for performing the operations of the methods. Such apparatus may be specially constructed for the required purposes, or may comprise a general purpose computer or other device selectively activated or reconfigured by a computer program stored in the computer. The algorithms and displays 20 presented herein are not inherently related to any particular computer or other apparatus.

Various general purpose machines may be used with programs in accordance with the teachings herein. Alternatively, the construction of more specialized apparatus to perform the required method steps may be appropriate. The structure of a conventional general purpose computer will appear from the description below.

25 In addition, the present specification also discloses a computer readable medium comprising a computer program for performing the operations of the methods. The computer readable medium is taken herein to include any transmission medium for communicating the computer program between a source and a designation. The transmission medium may include storage devices such as magnetic or optical disks, 564839.doc -11memory chips, or other storage devices suitable for interfacing with a general purpose computer. The transmission medium may also include a hard-wired medium such as exemplified in the Internet system, or wireless medium such as exemplified in the GSM mobile telephone system. The computer program is not intended to be limited to any particular programming language and implementation thereof. It will be appreciated that a variety of programming languages and coding thereof may be used to implement the teachings of the invention as described herein.

A method of determining the area enclosed by a planar curve is described below.

The method allows a user to set a desired upper limit on duration or lower limit on accuracy, when determining the area enclosed by the planar curve. In accordance with one arrangement, an initial approximation of the area enclosed by the planar curve is made, and an area estimate is then iteratively refined giving improved information with each pass.

For ease of explanation, the steps of the method described herein will be described with reference to a closed planar curve. However, the principles of the method described herein have general applicability to curve formats other than piece-wise cubic Bezier curves and it is not intended that the present invention be limited to the described methods. For example, the invention may have application to Hermite and Catmull-Rom curves.

S* 20 Fig. 16 is a flowchart showing a method of determining the area enclosed by a planar curve, in accordance with one arrangement. In this connection, Fig 1 shows a closed planar curve 101 enclosing an interior region 102. The area of the interior region 102 can be estimated using the method described herein. The curve 101 can be described as being composed of a number of cubic Bezier segments (eg. 205), as shown in Fig. 2.

25 The control polygon (eg. 201) for each Bezier segment includes "on-curve" points (eg.

202), indicated by circles, that correspond to the start and end points of each segment.

The control polygon 201 also includes "off-curve" points (eg. 203), indicated by squares, that correspond to interior control points within each segment's control polygon.

564839.doc -12- Fig. 3 shows a single cubic Bezier segment 309 and an associated control polygon 311. The start (Po) 301 and end (P 3 304 points are on-curve (ie. circles), and the interior control points (P 1 302 and (P 2 303 are off-curve points (ie. squares). A line 305 drawn from Po to P 3 describes a reference line around which the curve segment 309 encloses a particular area. In this case, the curve segment 309 is composed of a positive region 306 above the reference line 305, and negative region 307 below the reference line 305.

The process of Fig. 16 will now be explained with reference to the closed planar curve 101. The process of Fig. 16 begins at step 1601, where the on-curve polygon 402, as seen in Fig. 4, for the planar curve 101 is determined. At the next step 1603, the area of the on-curve polygon 402 is determined using any known polygonal area calculation method. The majority of known polygonal area calculation methods are relatively fast and exact. The polygonal area calculation method will be described in further detail later in this document. The process continues at the next step 1605, where the area (eg. 405) enclosed by each segment (eg. 407) relative to a reference line is determined. The method of determining the area (eg. 405) enclosed by each curve segment can include negative areas depending on the location of the segment with respect to the reference line, which will be described in more detail later in this document. The process concludes at the next step 1607, where the total area enclosed by the planar curve 101 is determined by S• 20 adding the area of the on-curve polygon 402, determined at step 1603, to the area enclosed by each segment relative to a reference line (NB: negative areas may be included), determined at step 1605.

The method of Fig. 16 will now be explained in more detail. Consider the on- *curve points (eg. 401) of the closed planar curve 101 in isolation, as shown in Fig 4. The 25 on-curve point polygon 402 encloses an area 403 that equates to a very rough approximation of the curve's 101 enclosed area. The area of the on-curve point polygon 402 provides a first guess at the area enclosed by the curve 101.

The polygonal area calculation method used to calculate the area of the polygon 402, will now be explained with reference to Fig 5. Firstly, a reference line 501 is drawn 564839.doc -13relative to an origin 502 in a direction 503. Secondly, the area of the quadrilaterals (eg. 504) formed by the projection of each polygonal line segment (eg. 505) onto the reference line 501 are added, or negated (ie. for line segments passing in the opposite direction) to calculate the area enclosed by polygon 402. The area 504 enclosed by the projection of the line segment (AB) 505 is shown shaded in Fig. 5. Alternatively, a person skilled in the art would recognise that any polygonal area calculation method can be used to calculate the area of the polygon 402.

After the on-curve polygon's 402 area has been calculated, the remaining area (ie. shown shaded in Fig. 6) described by the areas enclosed by each Bezier segment (eg.

601, 602) relative to their reference line is determined. The calculation of the remaining area can include both exterior regions (eg. 601) that are added to the total area or interior regions (eg. 602, 603) that are subtracted from the total area.

In contrast to the polygonal area calculation method discussed above, the calculation of the remaining area described by the areas enclosed by each Bezier segment, relative to a reference line, is not straightforward: Fig. 7 shows a curve segment 703 and a reference line 701. A control polygon 707, defined by the control points 709, 711, 713, 715, has been drawn for the curve segment 703. The area enclosed by the curve segment 703 and the reference line 701 can be approximated by determining the area enclosed by the control polygon 707 and the 20 reference line 701. Further, the area enclosed by the control polygon 707 and the reference line 701 can be approximated by calculating the area of the convex hull 702 for the control points 709, 711, 713, 715 (ie. the area shown shaded in Fig. 7).

The area enclosed by the curve segment 703 and the reference line 701 can be calculated more precisely by using subdivision at the midpoint 801 of the curve segment 703, as shown in Fig. 8, to produce two sub-curves 825 and 827. The control polygon 707 is re-drawn with reference to the midpoint 801 to produce two new control polygons 807 and 809. The control polygon 807 is defined by the control points 811, 813, 815 and 817. The control polygon 809 is defined by the control points 817, 819, 821 and 823.

However, note that the midpoint 801 does not necessarily lie on the reference line 701 of 564839.doc -14the curve 703. This means that new reference lines 806 and 805 for the two derived subcurves 825 and 827, respectively, do not necessarily lie on the reference line 701 of the curve 703. Using subdivision yields two more precise convex hull regions 802 and 803 (ie. shown shaded in Fig. which are defined by the control polygons 807 and 809, and the reference lines 806 and 805 respectively. The areas of the two convex hulls 802, 803 can be determined to calculate a more precise estimate of the area enclosed by the curve segment 703 and the reference line 701.

One disadvantage of using subdivision as discussed above, is that some gaps (eg.

829) may be introduced. These gaps may not be accounted for by either the on-curve polygonal area or the curve areas when calculating the area enclosed by the planar curve.

The area enclosed by the curve segment 703 and the reference line 701 can also be calculated by sampling the curve 703 at regular intervals and summing the positive (eg. 901) and negative (eg. 902) rectangular areas defined by the interval (eg. 903) of each sample, as shown in Fig. 9. Sampling the curve 703 at regular intervals can produce an even more precise approximation of the area enclosed by the Bezier curve segment 703 and the reference line 701, depending on the interval used. The smaller the interval 903, the more precise the estimation of the area enclosed by the curve segment 703 and the reference line 701.

The estimation of the area enclosed by the curve segment 703 and the reference 20 line 701, calculated by the method of Fig. 9, can be further refined using the method shown in Fig 10. Again, the curve 703 is sampled at regular intervals. However, a line SS. (eg. 1005) is then drawn between a point defined by the intersection of the curve 703 and

CCC.

one interval projection line (eg. 1007), and the intersection of the curve 703 and another interval projection line 1009. Triangular areas (eg. 1003) will be formed using the se.

C 25 method of Fig. 10. The area enclosed by the curve segment 703 and the reference line 701 is then calculated by summing the positive (eg. 1001) and negative (eg. 1002) polygonal areas defined by the interval of each sample.

Fig. 17 is a flowchart showing a method of determining the area enclosed by a curve segment (eg. 703) and a reference line (eg. 701). The method of Fig. 17 will be 564839.doc described with reference to the example curve segment 703 described along interval (t=0) 1106 to 1107, as shown in Fig. 11. In the method of Fig. 17, the method of Fig. is applied iteratively to give an improved estimate of the area enclosed by the curve segment 703 and the reference line 701, with each pass. The process of Fig. 17 begins at step 1701, where the curve 703 is sampled at its midpoint 1101 to obtain point P 1108. At the next step 1703, two sample areas 1102 and 1103 are defined (ie. triangles 1102 and 1103) by drawing a line from each of the end points 1106 and 1107 to the midpoint 1101, of the curve 703. The area represented by each of the triangles 1102 and 1103 can be determined and summed, at step 1703, to give an estimate of the area enclosed by the curve segment 703 and the reference line 701. However, the summing of the sample areas is preferably performed at the last iteration of the method of Fig. 17. At the next step 1705, if a predetermined time limit and/or a desired accuracy level limit has not been reached, then the process proceeds to step 1707, where the sampling interval is halved and the curve segment 703 is resampled using the new interval. Otherwise, the process concludes at step 1709, where the area represented by each of the sample areas i'°o (ie. triangles 1102 and 1103) can be determined and summed to give an estimate of the area enclosed by the curve segment 703 and the reference line 701.

oooo Continuing the example, during a second iteration of the method of Fig. 17, the curve segment 703 is sampled (step 1707) at the points (t=0.25) 1201 and (t=0.75) 1202 to obtain points Q 1203 and R 1204, as shown in Fig 12. Then, a line is drawn between each of the points 1106, 1203, 1108, 1204 and 1107 (step 1703), to define two quadrilaterals 1205, 1207 and two triangles 1209, 1211, as shown in Fig. 12. The area represented by each of the two quadrilaterals 1205, 1207 and two triangles 1209, 1211, o°,can be determined and summed to give an improved estimate of the area enclosed by the o£ curve segment 703 and the reference line 701. However, in accordance with the example, the process continues at next step 1705, where if the predetermined time limit and/or the desired accuracy level limit has not been reached, then the process proceeds to step 1707, where the curve segment 703 is sampled at the points (t=0.125) 1301, (t=0.375) 1302 (t=0.625) 1303 and (t=0.875) 1304 to obtain points S 1305, T 1306, U 1307, and V 1308, 564839.doc -16as shown in Fig 13. At the next step 1703, a line is drawn between each of the points 1106, 1305, 1203, 1306, 1108, 1307, 1204, 1308 and 1107, to define a set of quadrilaterals (eg. 1310) and triangles (eg. 1312). In accordance with the example, the d accuracy level limit has been reached and the process continues at the next step 1709, where the area represented by each of the quadrilaterals and triangles is determined and summed to give an estimate of the area enclosed by the curve segment 703 and the reference line 701. Therefore, the iterative process of Fig. 17 can be repeated until the specified time constraint or desired level of accuracy is reached.

The above method of determining the area enclosed by a planar curve was described with reference to a closed planar curve. However, in accordance with a further arrangement, the method of determining the area enclosed by a planar curve will now be described with reference to an open planar curve. Again, the method of the further arrangement allows a user to set a desired upper limit on duration or lower limit on o accuracy.

In accordance with the further arrangement, the area to be defined by the open curve is bounded by the end-points of the open curve. For example, Fig 14 shows an open planar curve 1401 described by a piecewise cubic Bezier curve and corresponding control polygon 1402. The curve 1401 has endpoints A 1403 and D 1404. A first approximation to the area enclosed by the curve 1401 can be determined by closing the curve across its endpoints with line AD 1501, as shown in Fig. 15, and calculating the oncurve polygonal area 1502. Note that no additional enclosed area needs be added for the segment described by line AD. The area enclosed by each of the curve segments (eg.

1503) and a reference line (eg. 1505), is then determined using the method of Fig. 17, with a user setting a desired upper limit on duration or lower limit on accuracy. The oncurve polygonal area 1502 is then added to the area enclosed by each of the curve segments, to determine the total area defined by the curve 1401.

Midpoint subdivision of Bezier curves is very efficient and therefore the aforementioned preferred methods are very suited to the recursive halving of the 't' intervals, as described above.

564839.doc -17- The aforementioned preferred methods comprise a particular control flow. There are many other variants of the preferred methods which use different control flows without departing the spirit or scope of the invention. Furthermore one or more of the steps of the preferred methods may be performed in parallel rather than sequentially.

The method of determining the area enclosed by a planar curve is preferably practiced using a conventional general-purpose computer system 1800, such as that shown in Fig. 18 wherein the processes of Figs. 1 to 17 may be implemented as software, such as an application program executing within the computer system 1800. In particular, the steps of method of determining the area enclosed by a planar curve are effected by instructions in the software that are carried out by the computer. The software may be divided into two separate parts; one part for carrying out the preferred methods; and another part to manage the user interface between the latter and the user. The software may be stored in a computer readable medium, including the storage devices described below, for example. The software is loaded into the computer from the computer 15 readable medium, and then executed by the computer. A computer readable medium o..

having such software or computer program recorded on it is a computer program product.

The use of the computer program product in the computer preferably effects an advantageous apparatus for determining the area enclosed by a planar curve.

The computer system 1800 comprises a computer module 1801, input devices such as a keyboard 1802 and mouse 1803, output devices including a printer 1815 and a display device 1814. A Modulator-Demodulator (Modem) transceiver device 1816 is used by the computer module 1801 for communicating to and from a communications network 1820, for example connectable via a telephone line 1821 or other functional medium. The modem 1816 can be used to obtain access to the Internet, and other network systems, such as a Local Area Network (LAN) or a Wide Area Network (WAN).

The computer module 1801 typically includes at least one processor unit 1805, a memory unit 1806, for example formed from semiconductor random access memory (RAM) and read only memory (ROM), input/output interfaces including a video interface 1807, and an 1/O interface 1813 for the keyboard 1802 and mouse 1803 and 564839.doc -18optionally a joystick (not illustrated), and an interface 1808 for the modem 1816. A storage device 1809 is provided and typically includes a hard disk drive 1810 and a floppy disk drive 1811. A magnetic tape drive (not illustrated) may also be used. A CD- ROM drive 1812 is typically provided as a non-volatile source of data. The components 1805 to 1813 of the computer module 1801, typically communicate via an interconnected bus 1804 and in a manner which results in a conventional mode of operation of the computer system 1800 known to those in the relevant art. Examples of computers on which the embodiments can be practised include IBM-PC's and compatibles, Sun Sparcstations or alike computer systems evolved therefrom.

Typically, the application program of the embodiments is resident on the hard disk drive 1810 and read and controlled in its execution by the processor 1805.

Intermediate storage of the program and any data fetched from the network 1820 may be accomplished using the semiconductor memory 1806, possibly in concert with the hard disk drive 1810. In some instances, the application program may be supplied to the user S15 encoded on a CD-ROM or floppy disk and read via the corresponding drive 1812 or 1811, .oo.

or alternatively may be read by the user from the network 1820 via the modem device 1816. Still further, the software can also be loaded into the computer system 1800 S* from other computer readable medium including magnetic tape, a ROM or integrated circuit, a magneto-optical disk, a radio or infra-red transmission channel between the 20 computer module 1801 and another device, a computer readable card such as a PCMCIA card, and the Internet and Intranets including email transmissions and information recorded on websites and the like. The foregoing is merely exemplary of relevant computer readable mediums. Other computer readable media may be practiced without departing from the scope and spirit of the invention.

The method of determining the area enclosed by a planar curve may alternatively be implemented in dedicated hardware such as one or more integrated circuits performing the functions or sub functions of Figs. 16 and 17. Such dedicated hardware may include graphic processors, digital signal processors, or one or more microprocessors and associated memories.

564839.doc 19- The foregoing describes only some embodiments of the present invention, and modifications and/or changes can be made thereto without departing from the scope and spirit of the invention, the embodiments being illustrative and not restrictive.

In the context of this specification, the word "comprising" means "including principally but not necessarily solely" or "having" or "including" and not "consisting only of'. Variations of the word comprising, such as "comprise" and "comprises" have corresponding meanings.

564839.doc

Claims (3)

1. 28. MAY. 2004 13: 37 SPRUSON 'FERGUSON NO. 5916 P. 6 Claims: 1. A method of displaying a planar curve image based on the rea enclosed by a planar curve defining the image, said method comprising the steps of: dividing said curve into at least one curve segment; dcternmining an area enclosed by a polygon based on said at least one curve segment; determining a remaining area enclosed by each of said curve segments, said remaining area being determined by iteratively sampling each of said curve segments to a predetermined interval to provide at least one sample region for each of said curve segments, and determining an area defined by each of said sample regions and a reference '0.line associated with each of said curve segments; :summing said area enclosed by. said polygon and said remaining rea to determie the area enclosed by said planar curve;, and displaying the planar curve image in accordance with the determined area enclosed by said planar curve. @020 2. The method according to claim 1, wherein said predetermined interval is dependent on a predetermined accracyleel 4* The method according to yoo claim s ,3 wherein said redetermin ed nerali said polygon is determined by projecting each edge of said polygon onto a line and adding or negating areas formed thereby. The method according to any one of claims 1 to 4, wherein said curve segment is a Bezier curve segment.
2. 564830.doc COMS ID No: SBMI-00769912 Received by IP Australia: Time 13:46 Date (Y-fv-d) 2004-05-28 28. MAY. 2004 13:37 SPRUSON FIERGUSON NO 96 P. 7 -21- 6. The method according to claim 5, wherein said remaining area enclosed by each of said curve segments is determined based on a convex hull formed by a control polygon associated with said segment. 7. The method according to claim 1, wherein said sample regions are summed at every iteration to provide an area estimate for each of said curve segments. 8. A method of displaying a planar curve image based on the area enclosed by a planar curve defining the image, said method comprising the steps of: dividing said curve into at least one curve segment; determining a polygon based on said at least one curve segment; deternining an area enclosed by said polygon; determining a remaining area enclosed by each of said curve segments, said 1 determining comprising for each said curve segment, the sub-steps of: (Da) sampling said curve segment at a predetermined interval to provide at least one sample region; deternining an area defined by each sample region and a reference line associated with each of said curve segments; (Dc) repeating steps (Da) and (Db) a predetermined number of times; (Dd) summing said areas defined by each sample region to provide said curve segment area; summing said area enclosed by said polygon and each of said curve segment areas to determine the area enclosed by said planar curve; and displaying the planar curve image in accordance with the determined area enclosed by said planar curve. 9. The method according to claim 8, wherein said predetermined number of times is dependent on a predetermined accuracy level.
3. 564839.de COMS ID No: SBMI-00769912 Received by IP Australia: Time (I-tm) 13:46 Date 2004-05-28 28. MAY. 2004 13:37 SPRUSON FERGUSON NO.5916 P. 8 -22- The method according to claim 8, wherein said predetermined number of times is dependent on a predetermined period. 11. The method according to any one of claims 8 to 10, wherein said area enclosed by said polygon is determined by projecting each edge of said polygon onto a line and adding or negating areas formed thereby. 12. The method according to any one of claims 8 to 11, wherein said curve segment is a Bezier curve segment 13. The method according to claim 12, wherein said remaining area enclosed by each of said curve segments is determined based on a convex hull formed by a control polygon associated with said segment. 14. The method according to claim 8, wherein said sample regions are summed at every iteration to provide an area estimate for each of said curve segments. 15. A method of displaying a planar curve image based on the area enclosed by a 20 planar curve defining the image, said method comprising the steps of: S(A) dividing said curve into at least one Bezier curve segment; determining a polygon based on said at least one Bezier curve segment; determining an area enclosed by said polygon; determining a remaining area enclosed by each of said Bezier curve segments, said determining comprising for each said Bezier curve segment, the sub-steps: (Da) sampling said curve segment at a predetermined interval to provide at least one sample region; (Db) determining an area defined by each sample region and an associated reference line; 564839.doc COMS ID No: SBMI-00769912 Received by IP Australia: Time 13:46 Date 2004-05-28 28. MAY. 2004 13:38 SPRUSON IFERGUSON NO.5916 P. 9 -23- (De) repeating steps (Da) and (Db) a predetermined number of times; (Dd) summing said areas defined by each sample region to provide said curve segment area; summing said area enclosed by said polygon and each of said Bezier curve segment areas to determine the area enclosed by said planar curve displaying the planar curve image in accordance with the determined area enclosed by said planar curve. 16. An apparatus for determining the area enclosed by a planar curve, said apparatus comprising: dividing means for dividing said curve into at least one curve segment; determining means for determining an area enclosed by a polygon based on said at least one curve segment, and for determining a remaining area enclosed by each of said :0 curve segments, said remaining area being determined by iteratively sampling each of said curve segments to a predetermined interval to provide at least one sample region for each of said curve segments, and determining an area defined by each of said sample regions and a reference line associated with each of said curve segments; and summing means for summing said area enclosed by said polygon and said remaining area to provide the area enclosed by said planar curve. :17. The apparatus according to claim 16, wherein said predetermined interval is dependent on a predetermined accuracy level. 18. The apparatus according to claim 16, wherein said predetermined interval is dependent on a predetermined period.- 19. The apparatus according to any one of claims 16 to 18, wherein said area enclosed by said polygon is determined by projecting each edge of said polygon onto a line and adding or negating areas formed thereby. 5M283§.doe COMS ID No: SBMI-00769912 Received by IP Australia: Time (1-tm) 13:46 Date (Y-N1-d) 2004-05-28 28. AY. 2004 13:38 SPRUSON IFERGUSON NO. 5916 P. -24- The apparatus according to any one of claims 16 to 18, wherein said curve segment is a Bezier curve segment. 21. The apparatus according to claim 20, wherein said remaining area enclosed by each of said curve segments is determined based on a convex hull formed by a control polygon associated with said segment. 22. The apparatus according to claim 16, wherein said sample regions are summied at every iteration to provide an area estimate for each of said curve segments. 23. An apparatus for determining the area enclosed by a planar curve, said apparatus comprising: means for dividing said curve into at least one curve segment; means for determining a polygon based on said at least one curve segment; means for determining an area cnclosed by said polygon; means for determining a remaining area enclosed by each of said curve segments, said means for determining further comprising: means for sampling each said curve segment at a predetermined :20 interval to provide at least one sample region; (ii) means for determining an area defined by each sample region and a reference line associated with each of said curve segments; (iii) means for summing said areas defined by each sample region to provide said curve segment area; and means for summing said area enclosed by said polygon and each of said curve segment areas to provide the area enclosed by said planar curve. 24. The apparatus according to claim 23, wherein said predetermined number of times is dependent on a predetermined accuracy level. 5648394dc COMS ID No: SBMI-00769912 Received by IF' Australia: Time 13:46 Date 2004-05-28 28. MAY. 2064 13:38 SPRUSON FERGUSON NO. 5916 P. 11 The apparatus according to claim 23, wherein said predetermined number of times is dependent on a predetermined period. 26. The apparatus according to any one of claims 23 to 25, wherein said area enclosed by said polygon is determiined by projecting each edge of said polygon onto a line and adding or negating areas formred thereby. 27. The apparatus according to any one of claims 23 to 26, wherein said curve segment is a Bezier curve segment. 28. The apparatus according to claim 27, wherein said remaining area enclosed by each of said curve segments is determined based on a convex hull formed by a control polygon associated with said segment. 29. The apparatus according to claim 23, wherein said sample regions are summed at 999eevery iteration to provide an area estimate for each of said curve -segments. :30. An apparatus for determining the area enclosed by a planar curve, said apparatus comprising: 0.0"0a memory for storing a program; a processor for executing the program, said program being configured to perform the following steps: dividing said curve into at [east one curve segment; determining an area enclosed by a polygon based on said at least one curve segment; deterining a remaining area enclosed by each of said curve segments, said remaining area being determined by iteratively sampling each of said curve segments to a predetermined interval to provide at least one sample region for 564839.dc COMS 10 No: SBMI-00769912 Received by IP Australia: Time 13:46 Date 2004-05-28 28. MAY. 2064 13:38 SPRUSON FERGUSON NO. 5916 P. 12 26- each of said curve segments, and determining an area defined by each of said sample regions and a reference line associated with each of said curve segments; and summing said area enclosed by said polygon and said remaining area to provide the area enclosed by said planar curve. 31. An apparatus for determiing the area enclosed by a planar curve, said apparatus comprising: a memory for storing a program; a processor for cxecuting the program, said program being configured to perform the following steps: dividing said curve into at least one curve segment; determining a polygon based on said at least one curve segment; determnining an area enclosed by said polygon; determining a remaining area enclosed by each of said curve segments, said determining comprising for each said carve segment, the sub-steps of: (Da) sampling said curve segment at a predetermined interval to provide at least one sample region; 20 (Db) determining an area defined by each sample region *and a reference line associated with each of said curve segments; (Dc) repeating steps (Da) and (Db) a predetermined number of times; (Dd) summing said areas defined by each sample region to provide said curve segment area; and summing said area enclosed by said polygon and each of said curve segment areas to provide the area enclosed by said planar curve. S(,4839.doc COMS ID No: SBMI-00769912 Received by IP Australia: Time 13:46 Date 20134-05-28 28. MAY. 2004 13:39 SPRUSON FERGUSON NO. 5916 P. 13 2+7- 32. An apparatus for determining the area enclosed by a planar curve, said apparatus comprising: a memory for storing a program; a processor for executing the program, said program being configured to perform the following steps: dividing said curve into at least one Bezier curve segment; determinig a polygon based on said at least one Bezier curve segment; detennining an area enclosed by said polygon; determining a remaining area enclosed by each of said Bezier curve segments, said determining comprising for each said Bezier curve segment, the sub- steps: 66 6l Sitr t (Da) sampling said curve segment at a predetermined interval to provide at least one sample region; (Db) determining an area defined by each sample region and an associated reference line; (Dc) repeating steps (Da) and (Db) a predetermined t number oftimes; (Dd) summing said areas defined by each sample region to e 4 e :e20 provide said curve segment area; and 4*C* O surruning said area enclosed by said polygon and each of said Bezier curve segment areasto provide the area enclosed by said planar curve. 33. A program for determining the area enclosed by a planar curve, said program comprising: code for dividing said curve into at least one curve segment; code for determining an area enclosed by a polygon based on said at least one curve segment; 564839.doc COMS ID No: SBMI-00769912 Received by IP Australia: Time 13:46 Date 2004-05-28 28. MAY. 2004 13:39 SPRUSON FERGUSON NO. 5916 P. 14 .28. code for determining a remaining area enclosed by each of said curve segments, said remaining area being determined by iteratively sampling each of said curve segments to a predetermined interval to provide at least one sample region for each of said curve segments, and determining an area defined by each of said sample regions and a reference line associated with each of said curve segments; and code for summing said area enclosed by said polygon and said remaining area to provide the area enclosed by said planar curve. 34. A program for determining the area enclosed by a planar curve, said program comprising code for performing the following steps: dividing said curve into at least one curve segment; determining a polygon based on said at least one curve segment; determining an area enclosed by said polygon; S" determining a remaining area enclosed by each of said curve segments, said 15 determining comprising for each said curve segment, the sub-steps of: (Da) sampling said curve segment at a predetermined interval to provide at least one sample region; (Db) determining an area defined by each sample region and a reference line associated with each of said curve segments; (Dc) repeating steps (Da) and (Db) a predetermined number of times; (Dd) summing said areas defined by each sample region to provide said .o curve segment area; and summing said area enclosed by said polygon and each of said curve segment areas to provide the area enclosed by said planar curve. A program for determining the area enclosed by a planar curve, said program comprising code for performing the following steps: dividing said curve into at least one Bezier curve segment; determining a polygon based on said at least one Bezier curve segment; 564839.doc COMS ID No: SBMI-00769912 Received by IP Australia: Time 13:46 Date 2004-05-28 28. MAY. 2004 13:39 SPRUSON FERGUSON NO. 59i6 P. 29 determining an area enclosed by said polygon; determining a remaining area enclosed by each of said Bezier curve segments, said determining comprising for each said Bezier curve segment, the sub-steps: (Da) sampling said curve segment at a predetermined interval to provide at least one sample region; (Db) determining an area defined by each sample region and an associated reference line; (Dc) repeating steps (Da) and (Db) a predetermined number of times; (Dd) summing said areas defined by each sample region to provide said curve segment area; and summing said area enclosed by said polygon and each of said Bezier curve segment areas to provide the area enclosed by said planar curve. 36. A program according to any one of claims 33, 34 or 35, wherein said program is stored in a memory medium of an apparatus and is executed by a processor of said apparatus. 37. A method of displaying a planar curve image based on the area enclosed by a planar curve defining the image, said method being substantially as herein described with o 20 reference to any one of the embodiments as illustrated in Figs. I to 17. 38. An apparatus for determining the area enclosed by a planar curve, substantially as herein described with reference to any one of the embodiments as illustrated in Figs. 1 to 17. 39. A program for determining the area enclosed by a planar curve, substantially as herein described with reference to any one of the embodiments as illustrated in Figs. I to 17. 564839.doc COMS ID No" SBMI-00769912 Received by IP Australia: Time 13:46 Date 2004-05-28 28. MAY. 2004 13:40 SPRUSON FERGUSON NO.5916 P. 16 DATED this the Fourteenth Day of August, 2001 Canon Kabushiki Kaisha Patent Attorneys for the Applicant SPRUSON FERGUSON *S @500 564839.doc COMS ID No: SBMI-00769912 Received by IP Australia: Time 13:46 Date 2004-05-28