tangency of conics and quadrics - wseasfor conics, and extend it to ﬁnd the tangency of quadrics...

Tangency of Conics and Quadrics

SUDANTHI N. R. WIJEWICKREMA, ANDREW P. PAPLINSKI CHARLES E. ESSONClayton School of Information Technology Colour Vision Systems Ltd

Monash University Bacchus MarshAUSTRALIA AUSTRALIA

{snw,app}@csse.monash.edu.au [email protected]

Abstract:- Our paper discusses a simple way of determining tangency of conics using the concept of pencilsof conics and the pole-polar relationship. We discuss the method, analyze the different situations of tangencyfor conics, and extend it to find the tangency of quadrics in 3d space. Although the basic theory behind it isknown [5], the novelty of the method lies in the efficient and robust way of solving the tangency problem andits 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 occurwhere the tangency of conics or quadrics shouldbe determined. For example, as explained in [8],in an automatic fruit grading application, the tan-gency between fruit and the rollers of a conveyorthey travel on, had to be found in the reconstructionof the fruit from stereo images in real-time. Themotivation for this paper was this application whichrequired fast and robust calculations as discussed insection 8, which led to a more generalized analysisof the tangency problem.

Usually, finding tangency is done by solving the twonon-linear conic/quadric equations and checking forcoincident roots. This results in relatively compli-cated quartic equations. Schneider and Eberly [4]discuss how the calculations could be simplified bytransforming one conic to the coordinate frame ofthe other but still we are left with quartics. The sit-uation becomes more complex when quadrics areinvolved. Here, the intersection is a curve and if

this tends to a point, the quadrics are tangent. Thedetermination of the intersection curve is a well re-searched problem (e.g., Wang et al. [3, 6, 7]).

These calculations may be cumbersome and are notsuitable for a real-time environment. We use insteadthe well known geometric concepts of pole-polar re-lationship between conics/quadrics for the determi-nation of tangency. This involves the solving of aneigensystem (3×3 for conics and4×4 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 andquadrics, while section 3 introduces the pole-polarrelationship. Section 4 discusses the tangency ofconics and section 5 analyzes the different cases.The results are extended to quadrics in 3d space insection 6. Sections 7 and 8 give a comparison ofmethods and a practical application respectively.

Proc. of the 6th WSEAS Int. Conf. on Signal Processing, Computational Geometry & Artificial Vision, Elounda, Greece, August 21-23, 2006 (pp21-29)

2 Conics and Quadrics

A conic is a curve in 2d and can be represented by3 × 3 symmetric matrix,C. A point x, given by ahomogeneous three element vector that satisfies eqn(1) lies on the conic.

xT Cx = 0 (1)

where,

C =[

C ccT c

], (2)

C is a2 × 2 symmetric matrix,c is a two elementvector andc is a scalar.

Similarly, a quadric is a surface in 3d, representedby a4 × 4 symmetric matrixQ. For a pointX rep-resented by a homogeneous four element vector, eqn(3) is satisfied.

XT QX = 0 (3)

where,

Q =[

Q qqT q

], (4)

Q is a3× 3 symmetric matrix,q is a three elementvector andq is a scalar.

3 Pole-Polar Relationship

Let us consider a conic in 2d space and any pointPin the plane of the conic but not on it (fig 1). LetAB andCD be any two lines throughP that cutthe conic atA, B, C andD respectively. Let theintersection ofAD andBC be denoted byQ andthat ofAC andBD by R. Then the linep goingthroughQ andR is unique for a given conic and

a pointP. Hence, the following theorem holds asshown by Young in [9].

P

Q

RB

A

C

D

Figure 1: Pole Polar Relationship of a Conic

Theorem 1: If P is a point in the plane of a conic,but not on the conic, there exists a uniquely deter-mined linep which contains the other two diagonalpoints of any complete quadrangle inscribed in theconic, one of whose diagonal points isP.

The linep thus uniquely defined by the pointP andthe conic is called thepolar of P with respect tothe conic. Then, the pointP is called thepoleof p.For a point on the conic, this quadrangle cannot beconstructed 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 triangleand their respective polars form its opposite sides, itis called aself-polar triangle. Then, the followingtheorem is satisfied.

Theorem 2: If A, B, C, D are four points on theconic, the diagonal trianglePQR of the quadrangleABCD is self-polar for the conic.

Proof of this theorem could be found in Semple andKneebone [5]. From this, we could deduce that, inthe system shown in fig 1,PQR forms a self-polartriangle.

The following theorems describe further character-istics of the pole-polar relationship of a conic. Proofand more details of these could be found in Baker[1], Semple and Kneebone [5] and Young [9].


Theorem 3: If P is a point on the conic, its tangentatP is the polar ofP with respect to the conic.

Theorem 4: The polar of a pointP with respectto a conic, passes through the points of contact ofthe tangents to the conic throughP, if such tangentsexist.

Theorem 5: As a pointQ moves on the polar of apointP, the polar ofQ rotates aboutP.

The pole-polar relationship with respect to a coniccan be represented conveniently in equation form asgiven in Hartley and Zisserman [2]. LetP be a pointrepresented by a homogeneous three element vectorandp be a similar representation of a line that satis-fiespTx = 0 for any pointx on it. If C is a conic,the pole-polar relationship is as in eqn (5).

p = CP (5)

4 Calculation of Common Poles

To find the relationship between two conics, we usethe pole-polar relationship. For example, if two con-ics are tangent to each other, they should share acommon tangent line which (from theorem 3) is es-sentially the polar of the point of contact. Hence,the common poles and polars of two conics provideus with useful information as to their relationship inspace.

Let the two conics beC1 andC2, and if there existsa common poleP and polarp, the following rela-tionship should be satisfied.

p = C1P

p = λC2P(6)

where,λ is a scalar parameter.

Adding the above equations, we get eqn (7) whichdenotes that the poleP and the polarp should be

common to any conic in the pencil of conics givenby C.

p = CP (7)

where,

C = C1 + λC2, (8)

and the scaling parameter ofp has been omitted.

The base conicsC1 andC2 of the pencil of conicsdenoted byC have either four points in common(distinct, common, real or complex) or an infinitenumber of common points. The latter case occursif the two base conics coincide or if they are de-generate conics with a line in common. For brevityof presentation, we do not consider the latter situa-tions.

Let us consider two base conics that have four pointsin common. Fig 3 shows this relationship. A quad-rangle can be drawn through the common pointsA,B, C andD and a self-polar trianglePQR can beobtained from their diagonals. This is common toeach member of the family of conics as they all gothrough the four common points.

Base Conics

R

Q

P

B

D

C

A

Figure 3: Self-Polar Triangle for a Pencil of Conics

Semple and Kneebone [5] show that there are justthree degenerate members in the pencil and thatthey are the line pairs(AB,CD), (AC,BD) and


−100 −50 0 50 100−120

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Coincident Poles

Non−CoincidentPolar

PolarsCoincident

PoleNon−Coincident

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(1c)

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(1d)−20 −10 0 10 20 30 40 50 60

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Self−Polar Triangle

P

Q

R

V

U

(1e)

Figure 2: Cases of Tangency

(AD,BC). This gives the common self-polar tri-angle which we use in our analysis. Subtracting theequations in eqn (6), we get(C1 − λC2)P = 0.This is a generalized eigenvalue problem which canbe solved to get the values ofλ andP. To simplifythis equation, we pre-multiply the above equationby the inverse ofC2, to obtain eqn (9).

(S − λI)P = 0 (9)

where,

S = C−12 C1, (10)

andI is the3× 3 identity matrix

The eigenvalues of this system give us the values forthe parameterλ and the eigenvectors give the com-mon polesP. The values ofP are then pluggedinto eqn (6) to get the corresponding polars. Notethat for this calculation, we considerC2 to be non-degenerate.

By examining the different properties of the self po-lar triangle thus obtained, we can determine the re-

lationship of two conics in space as discussed in thenext section.

5 Analysis of Different Cases

5.1 Tangency

If the two conics under consideration are tangent toeach other, they have a common polar which is tan-gent to both. That is, the quadrangle formed by thefour common points becomes a triangle as two ofthe points that form the quadrangle coincide. Thecommon tangent would pass at this point. Hence,a self polar triangle cannot be obtained in the caseof tangency. Here, we analyze the different cases oftangency and how to identify them using the char-acteristics 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 coin-ciding with each other, as illustrated in fig 2. Notethat the common poles and their corresponding po-


−30 −20 −10 0 10 20 30 40 50

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50

TriangleSelf−Polar

(2a)−20 0 20 40 60 80

−40

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−20

−10

0

10

20

30

40

50

Real Pole

Real Polar

(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 andintersect at another. Here, two of the poles obtainedare complex conjugate while the other is real. Thereal pole and polar give the common tangent pointand tangent line respectively. Since there is anothersituation where we get one real and two complexconjugate poles (Fig 4 (2b), where the conics inter-sect at two points) we need to distinguish betweenthe two cases.

For the two conics to be tangent, the real pole hasto lie on both of the conics as stated in theorem3. Hence, if the real pole is given byPr and sat-isfies either (or both) of the following conditions,they would be on the conics and hence they wouldbe tangent to each other.

PrT C1Pr = 0

PrT C2Pr = 0

(11)

Case (1e) is the only case of tangency where self-polar triangles could be drawn. The two conicstouch at two points (U and V) and the lineUVhas a unique poleP associated with it (from thedual of theorem 1). This pole-polar relationshipof P andUV is satisfied for both conics. Hence,tangents (PU andPV) could be drawn fromP tothese intersection points (theorem 4). If we chooseany point onUV, from theorem 5, we know thatits polar would go troughP. The conjugate of thispoint would be on the intersection of the two po-lars and form a self-polar triangle. More details and

proof could be found in Young [9] and Semple andKneebone [5]. Since we can find an infinite num-ber of such conjugate points, we get an infinity ofself-polar triangles for this system.

To distinguish this case of tangency from any othercase where self-polar triangles exist, we use the factthat one of the polar lines intersects the conic atthe tangency points. By solving for the intersectionpoints of the lines with one of the conics, and check-ing if they lie on the conic using eqn (11), we candetermine if the conics are tangent.

5.2 Non-Tangency

There are two cases of non-tangency: intersectionand separateness in space. Firstly, two conics couldintersect each other at four points as shown in fig4, case (2a) or two points as in case (2b). Case (2a)gives three real poles with one lying inside both con-ics and the other two lying outside. This situationgives similar results to case (3b) and we cannot dis-tinguish between the two cases just by analyzing thepoles. Case (2b) gives two complex poles and onereal pole, and the real pole lies outside the conics.

In the situation where the conics are disjoint inspace, we have two situations as shown in Fig 5 (3a)and (3b). Case (3b) gives similar results to (2b). Butcase (3a) can be easily identified by the fact that itgives three real poles and two of them lie inside eachof the conics while the other is outside both.


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(3a)

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40

60

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120

140

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Self−PolarTriangle

(3b)

Figure 5: Cases where the Conics are Separate

5.3 Degenerate Cases

As shown in eqn (10),C1 could be any conic whileC2 could only be non-degenerate. Here, we inves-tigate the situation whereC1 is degenerate. The in-stance of degeneracy may be parallel lines, coinci-dent lines, or intersecting lines. Similar to the caseof two non-degenerate conics, we observe that if theconics are tangent, two of the poles (and hence theircorresponding polars) coincide. Hence, we can usethe same method of calculation here as well. Fig 6illustrates this relationship.

−15 −10 −5 0 5 10 15

−10

−5

0

5

10

Parallel Lines

CoincidentPoles

CoincidentPolars

Figure 6: Degenerate Case

6 Extension to Quadrics

We use the same principles used in the case of con-ics and extend the method so that it could be appliedfor quadrics in 3d space. The pole-polar relationshipwith respect to a quadric is between a point (pole)and a plane (polar). For a pointP in 3d space repre-sented as a four element homogeneous vector, andthe quadricQ, we get the relationship in eqn (12),

whereP is the pole andπ is the polar plane as givenin Hartley and Zisserman [2].

π = QP (12)

For any point on the surface ofQ, the polar is thetangent plane to the quadric at that point. For twoquadrics to have common pole-polar pairs, eqn (13)should be satisfied.

π = Q1P

π = µQ2P(13)

where,Q1 andQ2 are the base quadrics andµ is ascalar parameter.

As discussed above, this results in a4 × 4 eigen-system, the solution of which gives us four pole-polar pairs. The concept of self-polar triangles isextended here to 3d where a self-polar tetrahedronis formed. Each pole in the self-polar system lies onthe polar planes of the other three poles. In otherwords, the pole is the intersection between the polarplanes of the other three poles in the system. Theyform the vertices of the tetrahedron while the polarplanes form its sides. Fig 7 illustrates this concept.


TetrahedronSelf−Polar

Figure 7: Self-Polar Tetrahedron

As in the case of conics, there are several situationsof tangency, intersection and separateness in spacefor two quadrics and could be analyzed in a similarmanner. Basically if the quadrics are only tangentto each other at one point, two of the poles coincideand lie on the surface of both. Fig 8 shows how thepoles and polar planes coincide in a typical case oftangency.

This method could be used for degenerate quadricssuch as cones and cylinders too. But since there isan inversion involved, one of them has to be non-degenerate. In the case of tangency, the coincidenceof two polars can be observed here as well.

Non−CoincidentPolars

Non−CoincidentPoles

Coincident Polars

CoincidentPoles

Figure 8: Pole-Polar Relationship at Tangency

7 Comparison

The method discussed is clearly simpler and fasterthan finding the intersection points between conicsand looking for coincident roots. The solution of a3 × 3 eigensystem is much simpler than that of thequartic equation that results in the solving of the twoconic equations [4]. This simplicity is more pro-nounced in the case of quadrics, where the calcula-tion is a4 × 4 eigensystem vs. a quartic with twovariables. Our method is clearly more efficient andsimpler in the determination of tangency.

Another way of calculating common poles whichis similar to the discussed method, is mentioned inSemple and Kneebone [5]. They state that the de-generate members of the family of conics given byeqn (8), can be found by equating to zero, the dis-criminant of the quadratic formC. These membersare intersecting line pairs that can be found by solv-ing the cubic equation thus obtained.(AB,CD),(AC,BD), and(AD,BC) given in fig 3 are thedegenerate members thus obtained. The intersec-tion points of these line pairs give the polesP, QandR which form the self polar triangle.

To check for tangency, the poles have to be found,and for this, the3 × 3 matrix of rank 2 that rep-resents the degenerate conic has to be converted toblock diagonal form. This transformation leads tothe determination of the intersection points (poles).

The first part of the calculation which is finding thethree degenerate conics is equivalent to the solvingof the3 × 3 eigensystem discussed in our method.Whereas in that, the poles which can be comparedto check for tangency were found straightaway, herewe need to perform an additional calculation to allthree conic matrices, to obtain the poles. Hence,it was observed that solving the eigensystem wasfaster than determining the degenerate members ofC and solving for the intersection points.

In the case of quadrics, the solution of a quar-tic equation is required to obtain the four coneswhich are the degenerate members of the familyof quadrics. The poles are found by determiningthe apices of the cones. Here too, this additional


calculation, as opposed to the direct calculation ofpoles from the eigensystem, makes it comparativelyslower.

When applied to conics (which were randomly gen-erated), the solving of the eigensystem was foundto be about75% faster on average, than the methodmentioned in Semple and Kneebone [5]. The sameapplies to the case of quadrics since the calculationof poles requires additional time. Our target appli-cation requires high speed and simple calculations,and hence, the method discussed in section 6 waschosen as the most suitable to determine tangencybetween two quadrics in 3d space.

8 Practical Application

Tangency of quadrics can be used in practical ap-plications such as that explained in [8], where ellip-soidal fruit such as citrus were modelled on rollers(cones). First, two images from either side of therollers were taken simultaneously as the fruit travelon the conveyor and ellipses were fitted to them.Then the fitted conics were adjusted to fit epipolartangency constraints. This is shown in fig 9.

50 100 150 200 250 300 350

0

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100

150

200

250

Fruit Fitted Ellipse

Rollers

Figure 9: View from Right Camera

A family of ellipsoids were created from these ad-justed conics and the ellipsoid that was tangent tothe rollers was chosen as the correct model. Tocheck for tangency of the fruit and the rollers, theprocess in section 6 was used. Fig 10 shows such areconstructed fruit on rollers.

Reconstructed Fruit

Rollers

Figure 10: Reconstruction

Experiments were conducted with ellipsoids ofknown dimensions and the average volume error forthe reconstruction was calculated. It was found thatthe average volume error was below4% and the av-erage reconstruction speed was approximately100ms when implemented in Matlab and run on a PCwith a CPU speed of 2.8 GHz. Some reconstructionerrors and the average are shown in table 1.

9 Conclusion

In this paper, we discussed how the tangency of twoconics in 2d space could be determined by solving a3× 3 eigensystem and observing the characteristicsof the results obtained. This method of calculationwas used in the case of one degenerate and one non-degenerate conic as well.

Then it was extended for quadrics in 3d space andthe tangency information was obtained by solvinga 4 × 4 eigensystem. The important point of themethod is that in situations of tangency, be it forconics or quadrics, degenerate or non-degenerate,two poles and their corresponding polars coincide.

Our observation of the method was that it was sim-ple and robust and was suitable for applications thathave requirements for high speed. It was success-fully used in such a real-life application, namely,the modelling of ellipsoidal fruit (such as citrus) onrollers to be used in the fruit grading industry [8].


Ellipsoid Reconstruction Error (%)

110x90x90 3.465390x75x75 3.167780x50x40 3.205860x50x40 3.870450x50x50 2.6390Average 3.4115

Table 1: Experimental Results

10 Acknowledgements

The authors would like to thank Dr. Ken Pledger,School of Mathematics, Statistics and ComputerScience, 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,1923.

[2] Richard Hartley and Andrew Zisserman.Mul-tiple View Geometry in Computer Vision. Cam-bridge University Press, 2003.

[3] S. Lazard, L. M. Pearanda, and S. Petitjean. In-

tersecting quadrics: An efficient and exact im-plementation. InProc. of SoCG (ACM Sympo-sium on Computational Geometry), pages 419–428, NewYork, USA, June 2004.

[4] Phillip J. Schneider and David H. Eberly.Geo-metric Tools for Computer Graphics. ElsevierScience (USA), 2003.

[5] J. G. Semple and G. T. Kneebone.AlgebraicProjective Geometry. Oxford University Press,1956.

[6] W. Wang, R. Goldman, and C. Tu. Enhancinglevin’s method for computing quadractic sur-face intersections.Computer Aided GeometricDesign, 20(7):401–422, 2003.

[7] W. Wang, B. Joe, and R. Goldman. Comput-ing quadractic surface intersections based on ananaylsis of plane cubic curves.Graphical Mod-els, 64(6):335–367, 2002.

[8] S. N. R. Wijewickrema, A. P. Paplinski, andC. E. Esson. Reconstruction of ellipsoids onrollers from stereo images using occluding con-tours. InInt. Conf. on Computer Vision Theoryand Applications, pages 370–377, Setubal, Por-tugal, February 2006.

[9] John Wesley Young.Projective Geometry. TheMathematical Association of America, 1930.


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