Conversion of Quadrics into Rational Biquadratic Bézier Patches

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1 Conversion of Quadrics into Rational Biquadratic Bézier Patches Lionel Garnier, Sebti Foufou, and Dominique Michelucci LEI, UMR CNRS 558 UFR Sciences, University of Burgundy, BP 7870, 078 Dijon Cedex, France <lgarnier,sfoufou, Abstract. The aim of this paper is to use symmetric properties of circles and Bernstein polynomials to define a series of interesting properties of rational biquadric Bézier patches, called barycentric properties. A robust algorithm based on these properties is proposed for the conrversion of revolution quadrics to rational biquadric Bézier surfaces. A set of conversion examples is given to illustrate the contribution of this algorithm. Introduction Rational Biquadratic Bézier Surfaces refereed to as RBBS in the rest of this paper are tensor product parametric surfaces widely used in the first generation of computer graphics applications and geometric modeling systems [, 5, 6]. Good introductions to RBBSs may be found in [6, 7,, 9]. Quadrics are second degree algebraic and parametric surfaces used as fundamental primitives in Boolean operations for solids algebra. The fact that quadrics can be represented by low degree implicit or parametric equations makes it possible to define quick and robust algorithms for the integration of these surfaces in computer graphics applications. Examples of research topics, related to quadrics, that gained big attentions are: Quadrics intersection [, 7,, 8], quadrics blending [, 9, 0, 3], the use of quadrics in 3D reconstruction and re-engineering [,, 0], quadrics in solid modeling [3, 8, 5]. The aim of this paper is to proof a series of useful properties of RBBSs, called barycentric properties, and to show how one can use these properties to convert revolution quadrics into RBBSs. Section gives a brief overview of revolution quadrics and rational quadric Bézier curves and surfaces. Section 3 shows the use of rational quadric Bézier curves to represent circular arcs. Section proposes the new barycentric properties of RBBSs. The quadrics to RBBSs conversion algorithm is presented in section 5. Section 6 gives some conversion examples. Conclusion and future extensions of this work are given in section 7.

2 Rational Quadric Bézier Curves and Surfaces A Rational Quadric Bézier Curves RQBC is a second degree parametric curve defined by: t w i B i t OP i, t [0; ] i0 w i B i t i0 where B i t are Bernstein polynomials defined as: B 0 t t, B t t t and B t t, for i {0,, }, and w i are weight associated to the control points P i. For a standard RQBC w 0 and w are equal to, while w can be used to control the type the conic defined by the curve. Rational Biquadratic Bézier Surfaces RBBS are defined by a tensor product of two RQBC by: u, v w ij B i u B j v More details on Bézier curves and surfaces can be found in [5, 9]. 3 Modeling Circular Arcs Using RQBC w ij B i u B j v OP ij RQBC can be used to represent conics. Three control points and a scalar value the weight of the middle control point are enough to define an arc of conic. In this section, we give some results on the expression of circular arcs using RQBC. Theorem shows how to define a circle from two points and two tangents on these points. Theorem presents how to compute the middle control point of the RQBC that represent a given circular arc. Theorem 3 shows how to compute the weight of the middle the middle control point of the RQBC that represent a given circular arc. Figure shows the modeling of circular arcs using RQBC. The geometric construction used for theorems, and 3 is as follows: CO 0, R is a circle of center O 0 and radius R. Segments [P 0 P ] and [P P ] are tangents to the circle at points P 0 and P. I is the midpoint of segment [P 0 P ]. P the median plane of segment [P 0 P ]. Theorem. Circle from two points and tangents at these points Circle CO 0, R exists if and only if P P and P / [P 0 P ]. The radius R O 0 P 0 and the center O 0 is given by formula: P 0 P P O 0 t 0P I t 0 I P 3 P 0 P

3 Fig.. Modeling circular arcs by RQBC In the plane determined by C, the geometric angle P 0 O 0 P is less than π. This means that if we take the parameterization γ of the circle in terms of cosine and sine such as P 0 γ θ 0, P γ θ, we have θ 0 θ < π. Theorem. Computing control point P when the center of the circle is known The RQBC is the arc of the circle C passing through P 0 and P. In this case, the control point P verifies: O 0 P 0 I P 0 I P t O 0 I t O 0 P 0 O 0 I Theorem 3. Computing the weight w. The RQBC defined by control points P 0, P and P and the weight w is a circular arc if and only if the following condition hold: The RQBC defines the small arc of circle if: It defines the big arc of circle if: + w O 0 I + w O 0 P 5 w O 0I R R O 0 P O 0I O 0 P 0 O 0 P 0 O 0 P > 0 6 w O 0I + R R + O 0 P O 0I + O 0 P 0 O 0 P 0 + O 0 P < 0 7 Proofs af these theorems can be easily obtained by combining properties of RQBC with those of the circle and scalar product. Barycentric Properties of RBBSs Let S 0 be the Rational Biquadric Bézier Surface RBBS defined according to formula by control points P ij 0 i,j and weights w ij 0 i,j with w 00

4 w 0 w 0 w. In order to represent surfaces with spherical curvatures by surface S 0, we should have the following constraints on control points: P 0 belongs to the median plane of [P 00 P 0 ], P 0 belongs to the median plane of [P 00 P 0 ], P belongs to the median plane of [P 0 P ] and P belongs to the median plane of [P 0 P ]. The following theorem introduces a series of interesting barycentric properties of RBBSs that helps in the expression of quadrics as RBBSs Theorem. Barycentric properties of RBBSs. Let I 0, J 0, I and J be respectively the midpoints of the segments [P 00 P 0 ], [P 00 P 0 ], [P 0 P ] and [P 0 P ]. We have the following four relations: 0,,, 0, OI0 + w 0 OP 0 + w 0 OI + w OP + w OJ0 + w 0 OP 0 + w 0 OJ + w OP + w Let G 0 be the isobarycenter of points P 00, P 0, P 0, P and G the barycenter of weighted points P 0, w 0, P 0, w 0, P, w, P, w. We define the value w w 0 + w 0 + w + w and G the barycenter of weighted points G 0, and G, w. The point M, verifies the two following formulas:, + w OG + w OP + w + w w M, P + w M, G 3 >From the latest formula we deduce that P belongs to the line M, G. 3. Let G 3 be the barycenter of weighted points P 00, 9, P 0, 9, P 0,, P,, P 0, 6w 0, P, 6w, P 0, 8w 0, P, w, and W 0 + 6w 0 + 8w 0 + w + 6w. The point M, verifies the two following formulas:, W OG 3 + w OP W + w W + w G 3 M, w G 3 P 5 >From the latest formula we deduce that P belongs to the line G 3 M,.

5 . Let G be the barycenter of weighted points P 00, 9, P 0,, P 0, 9, P,, P 0, 6w 0, P, 6w, P 0, 8w 0, P, w, and W 0 + 6w 0 + 8w 0 + w + 6w. The point M, verifies the two following formulas:, W OG + w OP W + w W + w G M, 6 w G P 7 >From the latest formula we deduce that P belongs to the line G M,. Proof.. In order to prove the expressions 8 and 9 for points 0, and,, let us recall that B 0 0, B 0 B 0 0, B, B B 0 0, B, B 0 B, and if I is the midpoint of segment [AB], then for every point O we have OA+ OB OI. By Formula, The point 0, on the RBBS is: 0, w 0j B j j0 + w 0 + w ij B i 0 B j w 0j B j j0 w 00 + w 0 + w 0 + w 0 w00 OP 0j w ij B i 0 B j OP ij OP 00 + w 0 OP 0 + w 0 OP 0 OP 00 + OP 0 + w 0 OP 0 OI 0 + w 0 OP 0 + w 0 OI0 + w 0 OP 0 + w 0 OI 0 + w 0OP 0

6 On the other hand, point, on the RBBS is:, w j B j j0 + w + w ij B i B j w j B j j0 w 0 + w + w + w w0 OP j w ij B i B j OP ij OP 0 + w OP + w OP OP 0 + OP + w OP OI + w OP + w OI + w OP + w OI + w OP Expressions 0 and of, 0 and, can be prooved in a similar way.. Proof of the expression for point, :, w ij B i B j w ij B i B j OP ij First, let us consider the denominator of this fraction: wi0 w ij B i B j B i + w i + w i w 00 + w 0 + w 0 + w 00 + w 0 + w 0 + w w 00 + w 0 + w 0 + w 6 + w + w 8 i0 w 0 + w + w + + w 0 + w 0 + w + w + w 0 + w 0 + w + w 8 w 0 + w + w + w + w

7 Second, in the same way the nominator can be expressed as: wi0 B i OP i0 + w i OP i + w i OP i i0 w 00 OP 00 + w 0 OP 0 + w 0 OP 0 + w 0 OP 0 + w OP + w OP w i0 OP 0 + w OP + w OP OP 00 + OP 0 + OP 0 + OP w0 OP 0 + w 0 OP 0 + w OP + w OP + w OP OG 0 + w OG + w OP 8 So we have:, +w+w OG 0 + w OG + w OP OG 0 + w OG + w OP + w + w 8 OG 0 + w OG + w OP + w + w + w OG + w OP + w + w >From these results, the proof of expression 3 is straightforward: + w + w M, M, + w M, G + w M, P So we have: w M, P + w M, G 3. Before starting the proof of expression for point,, recall that B 0 9 6, B 3 8 et B 6. By formula we have:, w ij B i B j w ij B i B j OP ij

8 The denominator can be easily reduced as: w ij B i B j B i i0 9wi w i 6 + w i 6 9w w 0 + w 0 + w + 6w 0 + 6w + 8w 0 + w + w w 0 + 6w + 8w 0 + w + w 6 W + w 6 using this result and developing the nominator lead to the intended proof:, 6 9 OP w 0 OP 0 + w 0 OP OP 0 + w OP + W + w w0 OP + 9 OP 0 + 6w OP + OP + W + w W OG 3 + w OP W + w >From these results we can deduce that : W + w G 3 M, w G 3 P which is the expression 5. The proof of expressions 6 and 7 can be obtained in the same way. 5 The Quadrics to RBBSs Conversion Algorithm This section proposes a new algorithm for converting a part of a revolution quadric into a rational biquadric Bézier patch. Some conversion illustrations are given. Terms quadric patch, sphere patch, cone patch or cylinder patch will be used to refer to the part of the quadric we are converting. A set of conversion illustrations is given in section 6. Following is the conversion algorithm: Given: A quadric surface defined by a parametric map Γ, and a patch on this quadric delimited by parameter values: θ 0, θ, ψ 0 and ψ with θ 0 θ < π. These values also define four circular curvature lines on the quadric surface Find: The representation of this part as a RBBS S over [0, ] defined by nine control points P ij and nine weights w ij, 0 i, j <. Weights w ij will be equal to one except w 0, w 0, w et w that can have positif or negative values. Proceeding:. Obtain corner control points directly by: P 00 Γ θ 0, ϕ 0, P 0 Γ θ, ϕ 0, P 0 Γ θ 0, ϕ, P Γ θ, ϕ. and ϕ 0 ϕ < π.

. Find centers of the four circles of curvature using equation 3 of theorem. 3. Find control points P 0, P 0, P and P on the median planes of curvature lines using equation of theorem.

Control points and the weight of γ u are P 00, P 0, P 0 and w 0. Those of γ v are P 00, P 0, P 0 and w 0. Find θ, solution of equation γ u Γ θ, ψ 0. Find ψ, solution of equation γ v Γ θ0, ψ.

Compute weight w, using equation 3 of theorem a b c d Fig.. Conversion of two sphere patches into standard RBBSs. a The first patch.

c The second patch, it is delimited by the north pole of the sphere. d The obtained equivalent RBBS. 6 Results on Quadric Patches Conversion 6.

9 . Find centers of the four circles of curvature using equation 3 of theorem. 3. Find control points P 0, P 0, P and P on the median planes of curvature lines using equation of theorem.. Calculate weights w 0, w 0, w and w using equations 6 and 7 of theorem Consider two RQBC γ u and γ v defined on the borders of the region to be converted. Control points and the weight of γ u are P 00, P 0, P 0 and w 0. Those of γ v are P 00, P 0, P 0 and w 0. Find θ, solution of equation γ u Γ θ, ψ 0. Find ψ, solution of equation γ v Γ θ0, ψ. Find ψ 3, solution of equation γ v Γ θ0, ψ Compute the last control point P as the intersection of lines G Γ θ, ψ and G 3 Γ θ, ψ 3. the construction of G and G 3 are given in theorem. 7. Compute weight w, using equation 3 of theorem a b c d Fig.. Conversion of two sphere patches into standard RBBSs. a The first patch. b The obtained equivalent RBBS, it is is positioned on the sphere to show that it represents exactly the same region as the converted patch. c The second patch, it is delimited by the north pole of the sphere. d The obtained equivalent RBBS. 6 Results on Quadric Patches Conversion 6. Conversion to Standard RBBSs Cases presented in this sub-section concern only the conversion of quadric patches into standard RBBSs where all weight associated to control points are positif. The conversion algorithm of section 5 is applied to determine control points as well as weights of the RBBS that represents the quadric patch. Conversion of Sphere Patches: Given a sphere S defined by the parametric map Γ as: [ Γ : [0; π] π ; π ] E 8 θ ψ Γ θ ψ R cos θ cos ψ, R sin θ cos ψ, R sin ψ

One can notice here that θ and ψ values needed to define lines containing γ u

Figure shows the conversion of two sphere patches into standard RBBSs.

needs a special consideration. For the north pole, we have ψ π and P 0 P P.

same as the RQBC defined by P 00, P 0 and P 0, so we must have w w 0.

A cone patch a and a cylinder patch c with their equivalent RBBS

Figure 3 shows a part of a cone and a part of a cylinder with their equivalent

We establish her that, in the case of the cone, it is possible to take the

10 One can notice here that θ and ψ values needed to define lines containing γ u control point P are given by: θ arctan y γ u and ψ γ ux arcsin z R. Figure shows the conversion of two sphere patches into standard RBBSs. The case of sphere patches delimited by one of the two poles of the sphere needs a special consideration. For the north pole, we have ψ π and P 0 P P. The RQBC defined by P 0, P and P is a point, but its nature should still the same as the RQBC defined by P 00, P 0 and P 0, so we must have w w 0. a b c d Fig. 3. A cone patch a and a cylinder patch c with their equivalent RBBS representations b and d respectively Conversion of Cone or Cylinder Patches: Figure 3 shows a part of a cone and a part of a cylinder with their equivalent RBBS representations. We establish her that, in the case of the cone, it is possible to take the vertex of the cone as a delimiter of the part to be converted. It is a similar situation to the pole of a sphere and we have: P 0 P P and w w 0. a b c Fig.. Conversion of a part of a cylinder a into two RBBSs: b is a standard RBBS, c is a RBBS with negative weights.

6. Conversion to RBBSs with Negative Weights By allowing negative weights for our RBBSs, we can see that it is also possible to convert more complex patches e.g. the whole cone or the whole cylinder.

Positive weights define the small patch subfigures b, while negative weights allows to define the large patch subfigures c.

Conversion of a piece of a cone a into two RBBSs: b is a standard RBBS, c is a RBBS with negative weights.

11 6. Conversion to RBBSs with Negative Weights By allowing negative weights for our RBBSs, we can see that it is also possible to convert more complex patches e.g. the whole cone or the whole cylinder. Figures, 5 show respectively conversions of larger patches of a cylinder and a cone into two RBBSs. For each case, the same nine control points and twelve weights define both of the obtained RBBSs. Positive weights define the small patch subfigures b, while negative weights allows to define the large patch subfigures c. In the two a subfigures, the obtained RBBSs are positioned on the quadric surface to show that they represent accurately the converted part. a b c Fig. 5. Conversion of a piece of a cone a into two RBBSs: b is a standard RBBS, c is a RBBS with negative weights. 7 Conclusion In this paper, we have proposed a theorem that describes a set of barycentric properties of rational biquadratic Bézier surfaces, thanks to the symmetry property of circles and Bernstein polynomials. A quadrics to RBBSs conversion algorithm based on these barycentric properties is also proposed. Several conversion experimental cases are given to show the results obtained by our conversion algorithm. References. C. Bajaj. Directions in Geometric Computing, R. Martin Editor, chapter The Emergence of Algebraic Curves and Surfaces in Geometric Design, pages 9. Information Geometers Press, G. Cross and A. Zisserman. Quadric surface reconstruction from dual-space geometry. In ICCV, pages 5 3, J. R. Davis, R. Nagel, and W. Guber. A model making and display technique for 3-d pictures. In Proceedings of the 7th Annual Meeting of UAIDE, pages 7 7, San Francisco, Oct G. Demengel and J. P. Pouget. Mathématiques des Courbes et des Surfaces. Modèles de Bézier, des B-Splines et des NURBS, volume. Ellipse, 998.

12 5. G. Farin. Curves and Surfaces for Computer Aided Geometric Design. Academic Press, San Diego, edition, J. Foley, A. Van Dam, D. Freiner, and J. Hughes. Computer Graphics : Principles and Practice. Addison Wesley, edition, A. Forest. Curves and Surfaces for Computer-Aided Design. PhD thesis, University of Cambridge, R. A. Goldstein and R. Nagel. 3d visual simulation. Simulation, pages 5 3, Jan J. Hoschek and D. Lasser. Fundamentals of Computer Aided Geometric Design. A.K.Peters, Wellesley, Massachussets, O. Jokinen. Reconstruction of quadric surfaces from disparity measurements. Applications of Digital Image Processing XVII Andrew G. Tescher, Proc. SPIE 98, San Diego, pages , 99.. Johnstone J. K.and Shene C. K. blending surfaces for cones. In Fisher R.B., editor, The Mathematics of Surfaces V, pages 3 9, Oxford, 99. Clarendon Press.. M. Kargerova. Velocity and coriolis quadrics of robot-manipulators. Mathematica Pannonica, 7: 5, Ku-Jin Kim, Myung-Soo Kim, and Kyungho Oh. Torus/sphere intersection based on a configuration space approach. Graphical models and image processing, 60:77 9, J. Z. Levin. Mathematical models for determining the intersections of quadric surfaces. Computer Vision, Graphics and Image Processing, pages 73 87, R. Mahl. Visible surface algorithms for quadric patches. Technical report UTECCSc-70-, CS Dept., University of Utah, L. Piegl and W. Tilles. A managerie of rational b-spline circles. IEEE Computer Graphics and Applications, 95:6 56, R. Sarraga. Algebraic methods for intersections of quadric surfaces in gmsolid. Technical Report GMR-39, Computer Science Departement, General Motors Research Labs, Ching-Kuang Shene and John K. Johnstone. On the lower degree intersections of two natural quadrics. ACM Transactions on Graphics, 3:00, October J. Wallner. Geometric Contributions to Surface Modeling. PhD Thesis, Vienna University of Technology, J. Wallner and H. Pottmann. Rational blending surfaces between quadrics. Computer Aided Geometric Design, 5:07 9, P. Y. Woon and H. Freeman. A computer procedure for generating visible-line projections of solids bounded by quadric surfaces. In Information Processing 7, volume, pages 0 5, Amsterdam, 97. North-Holland Publishing Co.

Geometric meanings of the parameters on rational conic segments

Science in China Ser. A Mathematics 005 Vol.48 No.9 09 09 Geometric meanings of the parameters on rational conic segments HU Qianqian & WANG Guojin Department of Mathematics, Zhejiang University, Hangzhou