Tangency of Conics and Quadrics

Transcription

1 Proc. of the 6th WSEAS Int. Conf. on Signal Processing, Computational Geometry & Artificial Vision, Elounda, Greece, August 21-23, 26 (pp21-29) Tangency of Conics and Quadrics SUDANTHI N. R. WIJEWICKREMA, ANDREW P. PAPLIŃSKI Clayton School of Information Technology Monash University AUSTRALIA CHARLES E. ESSON Colour Vision Systems Ltd Bacchus Marsh AUSTRALIA Abstract: - Our paper discusses a simple way of determining tangency of conics using the concept of pencils of conics and the pole-polar relationship. We discuss the method, analyze the different situations of tangency for conics, and extend it to find the tangency of quadrics in 3d space. Although the basic theory behind it is known [5], the novelty of the method lies in the efficient and robust way of solving the tangency problem and its successful application to a real-life problem: namely, the modelling of fruit on rollers for fruit grading. The simplicity of the calculation makes it attractive for applications where speed is of importance. Key-Words: - Tangency, Intersection, Conics, Quadrics, Pole-Polar Relationship 1 Introduction In computer vision and graphics, situations occur where the tangency of conics or quadrics should be determined. For example, as explained in [8], in an automatic fruit grading application, the tangency between fruit and the rollers of a conveyor they travel on, had to be found in the reconstruction of the fruit from stereo images in real-time. The motivation for this paper was this application which required fast and robust calculations as discussed in section 8, which led to a more generalized analysis of the tangency problem. Usually, finding tangency is done by solving the two non-linear conic/quadric equations and checking for coincident roots. This results in relatively complicated quartic equations. Schneider and Eberly [4] discuss how the calculations could be simplified by transforming one conic to the coordinate frame of the other but still we are left with quartics. The situation becomes more complex when quadrics are involved. Here, the intersection is a curve and if this tends to a point, the quadrics are tangent. The determination of the intersection curve is a well researched problem (e.g., Wang et al. [3, 6, 7]). These calculations may be cumbersome and are not suitable for a real-time environment. We use instead the well known geometric concepts of pole-polar relationship between conics/quadrics for the determination of tangency. This involves the solving of an eigensystem (3 3 for conics and 4 4 for quadrics), which is superior computationally and numerically, and hence is more suitable for the target application. The rest of the paper is organized as follows: Section 2 summarizes the concepts of conics and quadrics, while section 3 introduces the pole-polar relationship. Section 4 discusses the tangency of conics and section 5 analyzes the different cases. The results are extended to quadrics in 3d space in section 6. Sections 7 and 8 give a comparison of methods and a practical application respectively.

2 Proc. of the 6th WSEAS Int. Conf. on Signal Processing, Computational Geometry & Artificial Vision, Elounda, Greece, August 21-23, 26 (pp21-29) 2 Conics and Quadrics a point P. Hence, the following theorem holds as shown by Young in [9]. A conic is a curve in 2d and can be represented by 3 3 symmetric matrix, C. A point x, given by a homogeneous three element vector that satisfies eqn (1) lies on the conic. P C x T Cx = (1) A Q D where, R B [ C c C = c T c ], (2) Figure 1: Pole Polar Relationship of a Conic C is a 2 2 symmetric matrix, c is a two element vector and c is a scalar. Similarly, a quadric is a surface in 3d, represented by a 4 4 symmetric matrix Q. For a point X represented by a homogeneous four element vector, eqn (3) is satisfied. where, X T QX = (3) [ Q q Q = q T q ], (4) Q is a 3 3 symmetric matrix, q is a three element vector and q is a scalar. 3 Pole-Polar Relationship Let us consider a conic in 2d space and any point P in the plane of the conic but not on it (fig 1). Let AB and CD be any two lines through P that cut the conic at A, B, C and D respectively. Let the intersection of AD and BC be denoted by Q and that of AC and BD by R. Then the line p going through Q and R is unique for a given conic and Theorem 1: If P is a point in the plane of a conic, but not on the conic, there exists a uniquely determined line p which contains the other two diagonal points of any complete quadrangle inscribed in the conic, one of whose diagonal points is P. The line p thus uniquely defined by the point P and the conic is called the polar of P with respect to the conic. Then, the point P is called the pole of p. For a point on the conic, this quadrangle cannot be constructed and the relationship between the conic, pole and polar is as given in theorem 3 below. If the poles of a conic form the vertices of a triangle and their respective polars form its opposite sides, it is called a self-polar triangle. Then, the following theorem is satisfied. Theorem 2: If A, B, C, D are four points on the conic, the diagonal triangle PQR of the quadrangle ABCD is self-polar for the conic. Proof of this theorem could be found in Semple and Kneebone [5]. From this, we could deduce that, in the system shown in fig 1, PQR forms a self-polar triangle. The following theorems describe further characteristics of the pole-polar relationship of a conic. Proof and more details of these could be found in Baker [1], Semple and Kneebone [5] and Young [9].

3 Proc. of the 6th WSEAS Int. Conf. on Signal Processing, Computational Geometry & Artificial Vision, Elounda, Greece, August 21-23, 26 (pp21-29) Theorem 3: If P is a point on the conic, its tangent at P is the polar of P with respect to the conic. common to any conic in the pencil of conics given by C. Theorem 4: The polar of a point P with respect to a conic, passes through the points of contact of the tangents to the conic through P, if such tangents exist. where, p = CP (7) Theorem 5: As a point Q moves on the polar of a point P, the polar of Q rotates about P. C = C 1 + λc 2, (8) The pole-polar relationship with respect to a conic can be represented conveniently in equation form as given in Hartley and Zisserman [2]. Let P be a point represented by a homogeneous three element vector and p be a similar representation of a line that satisfies p T x = for any point x on it. If C is a conic, the pole-polar relationship is as in eqn (5). p = CP (5) 4 Calculation of Common To find the relationship between two conics, we use the pole-polar relationship. For example, if two conics are tangent to each other, they should share a common tangent line which (from theorem 3) is essentially the polar of the point of contact. Hence, the common poles and polars of two conics provide us with useful information as to their relationship in space. Let the two conics be C 1 and C 2, and if there exists a common pole P and polar p, the following relationship should be satisfied. and the scaling parameter of p has been omitted. The base conics C 1 and C 2 of the pencil of conics denoted by C have either four points in common (distinct, common, real or complex) or an infinite number of common points. The latter case occurs if the two base conics coincide or if they are degenerate conics with a line in common. For brevity of presentation, we do not consider the latter situations. Let us consider two base conics that have four points in common. Fig 3 shows this relationship. A quadrangle can be drawn through the common points A, B, C and D and a self-polar triangle PQR can be obtained from their diagonals. This is common to each member of the family of conics as they all go through the four common points. B Q A Base Conics C P D p = C 1 P p = λc 2 P (6) Figure 3: Self-Polar Triangle for a Pencil of Conics R where, λ is a scalar parameter. Adding the above equations, we get eqn (7) which denotes that the pole P and the polar p should be Semple and Kneebone [5] show that there are just three degenerate members in the pencil and that they are the line pairs (AB, CD), (AC, BD) and

4 Proc. of the 6th WSEAS Int. Conf. on Signal Processing, Computational Geometry & Artificial Vision, Elounda, Greece, August 21-23, 26 (pp21-29) Non Polar Non Pole Non Polar Non Pole 6 Non Polar Non Pole 12 (1a) (1b) (1c) P R Triangle Q U 2 1 Real Pole Real Polar 1 V (1d) (1e) Figure 2: Cases of Tangency (AD, BC). This gives the common self-polar triangle which we use in our analysis. Subtracting the equations in eqn (6), we get (C 1 λc 2 )P =. This is a generalized eigenvalue problem which can be solved to get the values of λ and P. To simplify this equation, we pre-multiply the above equation by the inverse of C 2, to obtain eqn (9). where, and I is the 3 3 identity matrix (S λi)p = (9) S = C 1 2 C 1, (1) The eigenvalues of this system give us the values for the parameter λ and the eigenvectors give the common poles P. The values of P are then plugged into eqn (6) to get the corresponding polars. Note that for this calculation, we consider C 2 to be nondegenerate. By examining the different properties of the self polar triangle thus obtained, we can determine the relationship of two conics in space as discussed in the next section. 5 Analysis of Different Cases 5.1 Tangency If the two conics under consideration are tangent to each other, they have a common polar which is tangent to both. That is, the quadrangle formed by the four common points becomes a triangle as two of the points that form the quadrangle coincide. The common tangent would pass at this point. Hence, a self polar triangle cannot be obtained in the case of tangency. Here, we analyze the different cases of tangency and how to identify them using the characteristics of the common poles and polars. The first three cases shown in fig 2, i.e. (1a), (1b) and (1c), are the most common cases of tangency. Here, out of the three common poles, two coincide. This results in the two corresponding polars coinciding with each other, as illustrated in fig 2. Note that the common poles and their corresponding po-

5 Proc. of the 6th WSEAS Int. Conf. on Signal Processing, Computational Geometry & Artificial Vision, Elounda, Greece, August 21-23, 26 (pp21-29) Triangle Real Polar Real Pole (2a) (2b) Figure 4: Cases where the Conics Intersect lars are shown in the same color. In case (1d), the conics are tangent at one point and intersect at another. Here, two of the poles obtained are complex conjugate while the other is real. The real pole and polar give the common tangent point and tangent line respectively. Since there is another situation where we get one real and two complex conjugate poles (Fig 4 (2b), where the conics intersect at two points) we need to distinguish between the two cases. For the two conics to be tangent, the real pole has to lie on both of the conics as stated in theorem 3. Hence, if the real pole is given by P r and satisfies either (or both) of the following conditions, they would be on the conics and hence they would be tangent to each other. proof could be found in Young [9] and Semple and Kneebone [5]. Since we can find an infinite number of such conjugate points, we get an infinity of self-polar triangles for this system. To distinguish this case of tangency from any other case where self-polar triangles exist, we use the fact that one of the polar lines intersects the conic at the tangency points. By solving for the intersection points of the lines with one of the conics, and checking if they lie on the conic using eqn (11), we can determine if the conics are tangent. 5.2 Non-Tangency P r T C 1 P r = P r T C 2 P r = (11) Case (1e) is the only case of tangency where selfpolar triangles could be drawn. The two conics touch at two points (U and V) and the line UV has a unique pole P associated with it (from the dual of theorem 1). This pole-polar relationship of P and UV is satisfied for both conics. Hence, tangents (PU and PV) could be drawn from P to these intersection points (theorem 4). If we choose any point on UV, from theorem 5, we know that its polar would go trough P. The conjugate of this point would be on the intersection of the two polars and form a self-polar triangle. More details and There are two cases of non-tangency: intersection and separateness in space. Firstly, two conics could intersect each other at four points as shown in fig 4, case (2a) or two points as in case (2b). Case (2a) gives three real poles with one lying inside both conics and the other two lying outside. This situation gives similar results to case (3b) and we cannot distinguish between the two cases just by analyzing the poles. Case (2b) gives two complex poles and one real pole, and the real pole lies outside the conics. In the situation where the conics are disjoint in space, we have two situations as shown in Fig 5 (3a) and (3b). Case (3b) gives similar results to (2b). But case (3a) can be easily identified by the fact that it gives three real poles and two of them lie inside each of the conics while the other is outside both.

6 Proc. of the 6th WSEAS Int. Conf. on Signal Processing, Computational Geometry & Artificial Vision, Elounda, Greece, August 21-23, 26 (pp21-29) Triangle Triangle (3a) (3b) Figure 5: Cases where the Conics are Separate 5.3 Degenerate Cases As shown in eqn (1), C 1 could be any conic while C 2 could only be non-degenerate. Here, we investigate the situation where C 1 is degenerate. The instance of degeneracy may be parallel lines, coincident lines, or intersecting lines. Similar to the case of two non-degenerate conics, we observe that if the conics are tangent, two of the poles (and hence their corresponding polars) coincide. Hence, we can use the same method of calculation here as well. Fig 6 illustrates this relationship. where P is the pole and π is the polar plane as given in Hartley and Zisserman [2]. π = QP (12) For any point on the surface of Q, the polar is the tangent plane to the quadric at that point. For two quadrics to have common pole-polar pairs, eqn (13) should be satisfied Parallel Lines π = Q 1 P π = µq 2 P (13) Figure 6: Degenerate Case where, Q 1 and Q 2 are the base quadrics and µ is a scalar parameter. 6 Extension to Quadrics We use the same principles used in the case of conics and extend the method so that it could be applied for quadrics in 3d space. The pole-polar relationship with respect to a quadric is between a point (pole) and a plane (polar). For a point P in 3d space represented as a four element homogeneous vector, and the quadric Q, we get the relationship in eqn (12), As discussed above, this results in a 4 4 eigensystem, the solution of which gives us four polepolar pairs. The concept of self-polar triangles is extended here to 3d where a self-polar tetrahedron is formed. Each pole in the self-polar system lies on the polar planes of the other three poles. In other words, the pole is the intersection between the polar planes of the other three poles in the system. They form the vertices of the tetrahedron while the polar planes form its sides. Fig 7 illustrates this concept.

7 Proc. of the 6th WSEAS Int. Conf. on Signal Processing, Computational Geometry & Artificial Vision, Elounda, Greece, August 21-23, 26 (pp21-29) 7 Comparison Tetrahedron Figure 7: Self-Polar Tetrahedron As in the case of conics, there are several situations of tangency, intersection and separateness in space for two quadrics and could be analyzed in a similar manner. Basically if the quadrics are only tangent to each other at one point, two of the poles coincide and lie on the surface of both. Fig 8 shows how the poles and polar planes coincide in a typical case of tangency. This method could be used for degenerate quadrics such as cones and cylinders too. But since there is an inversion involved, one of them has to be nondegenerate. In the case of tangency, the coincidence of two polars can be observed here as well. Non Non Figure 8: Pole-Polar Relationship at Tangency The method discussed is clearly simpler and faster than finding the intersection points between conics and looking for coincident roots. The solution of a 3 3 eigensystem is much simpler than that of the quartic equation that results in the solving of the two conic equations [4]. This simplicity is more pronounced in the case of quadrics, where the calculation is a 4 4 eigensystem vs. a quartic with two variables. Our method is clearly more efficient and simpler in the determination of tangency. Another way of calculating common poles which is similar to the discussed method, is mentioned in Semple and Kneebone [5]. They state that the degenerate members of the family of conics given by eqn (8), can be found by equating to zero, the discriminant of the quadratic form C. These members are intersecting line pairs that can be found by solving the cubic equation thus obtained. (AB, CD), (AC, BD), and (AD, BC) given in fig 3 are the degenerate members thus obtained. The intersection points of these line pairs give the poles P, Q and R which form the self polar triangle. To check for tangency, the poles have to be found, and for this, the 3 3 matrix of rank 2 that represents the degenerate conic has to be converted to block diagonal form. This transformation leads to the determination of the intersection points (poles). The first part of the calculation which is finding the three degenerate conics is equivalent to the solving of the 3 3 eigensystem discussed in our method. Whereas in that, the poles which can be compared to check for tangency were found straightaway, here we need to perform an additional calculation to all three conic matrices, to obtain the poles. Hence, it was observed that solving the eigensystem was faster than determining the degenerate members of C and solving for the intersection points. In the case of quadrics, the solution of a quartic equation is required to obtain the four cones which are the degenerate members of the family of quadrics. The poles are found by determining the apices of the cones. Here too, this additional

8 Proc. of the 6th WSEAS Int. Conf. on Signal Processing, Computational Geometry & Artificial Vision, Elounda, Greece, August 21-23, 26 (pp21-29) calculation, as opposed to the direct calculation of poles from the eigensystem, makes it comparatively slower. When applied to conics (which were randomly generated), the solving of the eigensystem was found to be about 75% faster on average, than the method mentioned in Semple and Kneebone [5]. The same applies to the case of quadrics since the calculation of poles requires additional time. Our target application requires high speed and simple calculations, and hence, the method discussed in section 6 was chosen as the most suitable to determine tangency between two quadrics in 3d space. 8 Practical Application Tangency of quadrics can be used in practical applications such as that explained in [8], where ellipsoidal fruit such as citrus were modelled on rollers (cones). First, two images from either side of the rollers were taken simultaneously as the fruit travel on the conveyor and ellipses were fitted to them. Then the fitted conics were adjusted to fit epipolar tangency constraints. This is shown in fig 9. Rollers Reconstructed Fruit Figure 1: Reconstruction Experiments were conducted with ellipsoids of known dimensions and the average volume error for the reconstruction was calculated. It was found that the average volume error was below 4% and the average reconstruction speed was approximately 1 ms when implemented in Matlab and run on a PC with a CPU speed of 2.8 GHz. Some reconstruction errors and the average are shown in table 1. 9 Conclusion Fruit Fitted Ellipse In this paper, we discussed how the tangency of two conics in 2d space could be determined by solving a 3 3 eigensystem and observing the characteristics of the results obtained. This method of calculation was used in the case of one degenerate and one nondegenerate conic as well. 2 Rollers Figure 9: View from Right Camera Then it was extended for quadrics in 3d space and the tangency information was obtained by solving a 4 4 eigensystem. The important point of the method is that in situations of tangency, be it for conics or quadrics, degenerate or non-degenerate, two poles and their corresponding polars coincide. A family of ellipsoids were created from these adjusted conics and the ellipsoid that was tangent to the rollers was chosen as the correct model. To check for tangency of the fruit and the rollers, the process in section 6 was used. Fig 1 shows such a reconstructed fruit on rollers. Our observation of the method was that it was simple and robust and was suitable for applications that have requirements for high speed. It was successfully used in such a real-life application, namely, the modelling of ellipsoidal fruit (such as citrus) on rollers to be used in the fruit grading industry [8].

9 Proc. of the 6th WSEAS Int. Conf. on Signal Processing, Computational Geometry & Artificial Vision, Elounda, Greece, August 21-23, 26 (pp21-29) Ellipsoid Reconstruction Error (%) 11x9x x75x x5x x5x x5x Average Table 1: Experimental Results 1 Acknowledgements The authors would like to thank Dr. Ken Pledger, School of Mathematics, Statistics and Computer Science, Victoria University, New Zealand and Mr. Daniel Tokarev, School of Mathematical Sciences, Monash University, Australia for their assistance. References: [1] H. F. Baker. Principles of Geometry. Volume 3, Solid Geometry. Cambridge University Press, [2] Richard Hartley and Andrew Zisserman. Multiple View Geometry in Computer Vision. Cambridge University Press, 23. [3] S. Lazard, L. M. Pearanda, and S. Petitjean. Intersecting quadrics: An efficient and exact implementation. In Proc. of SoCG (ACM Symposium on Computational Geometry), pages , NewYork, USA, June 24. [4] Phillip J. Schneider and David H. Eberly. Geometric Tools for Computer Graphics. Elsevier Science (USA), 23. [5] J. G. Semple and G. T. Kneebone. Algebraic Projective Geometry. Oxford University Press, [6] W. Wang, R. Goldman, and C. Tu. Enhancing levin s method for computing quadractic surface intersections. Computer Aided Geometric Design, 2(7):41 422, 23. [7] W. Wang, B. Joe, and R. Goldman. Computing quadractic surface intersections based on an anaylsis of plane cubic curves. Graphical Models, 64(6): , 22. [8] S. N. R. Wijewickrema, A. P. Papliński, and C. E. Esson. Reconstruction of ellipsoids on rollers from stereo images using occluding contours. In Int. Conf. on Computer Vision Theory and Applications, pages , Setubal, Portugal, February 26. [9] John Wesley Young. Projective Geometry. The Mathematical Association of America, 193.