conversion of quadrics into rational biquadratic bézier patches
TRANSCRIPT
Conversion of Quadrics into RationalBiquadratic Bezier Patches
Lionel Garnier, Sebti Foufou, and Dominique Michelucci
LE2I, UMR CNRS 5158UFR Sciences, University of Burgundy, BP 47870,
21078 Dijon Cedex, France<lgarnier,sfoufou, dmichel>@u-bourgogne.fr
Abstract. The aim of this paper is to use symmetric properties of cir-cles and Bernstein polynomials to define a series of interesting propertiesof rational biquadric Bezier patches, called barycentric properties. A ro-bust algorithm based on these properties is proposed for the conrversionof revolution quadrics to rational biquadric Bezier surfaces. A set of con-version examples is given to illustrate the contribution of this algorithm.
1 Introduction
Rational Biquadratic Bezier Surfaces (refereed to as RBBS in the rest of thispaper) are tensor product parametric surfaces widely used in the first genera-tion of computer graphics applications and geometric modeling systems [1, 5, 6].Good introductions to RBBSs may be found in [16, 7, 4, 9].
Quadrics are second degree algebraic (and parametric) surfaces used as fun-damental primitives in Boolean operations for solids algebra. The fact thatquadrics can be represented by low degree implicit or parametric equationsmakes it possible to define quick and robust algorithms for the integration ofthese surfaces in computer graphics applications. Examples of research topics,related to quadrics, that gained big attentions are: Quadrics intersection [14, 17,21, 18], quadrics blending [11, 19, 20, 13], the use of quadrics in 3D reconstructionand re-engineering [12, 2, 10], quadrics in solid modeling [3, 8, 15].
The aim of this paper is to proof a series of useful properties of RBBSs, calledbarycentric properties, and to show how one can use these properties to convertrevolution quadrics into RBBSs. Section 2 gives a brief overview of revolutionquadrics and rational quadric Bezier curves and surfaces. Section 3 shows the useof rational quadric Bezier curves to represent circular arcs. Section 4 proposesthe new barycentric properties of RBBSs. The quadrics to RBBSs conversionalgorithm 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 Bezier Curves and Surfaces
A Rational Quadric Bezier Curves (RQBC) is a second degree parametric curvedefined by:
−−−−→OM(t) =
12∑
i=0
wiBi(t)
(2∑
i=0
wiBi(t)−−→OPi
), t ∈ [0; 1] (1)
where Bi(t) are Bernstein polynomials defined as: B0(t) = (1− t)2 , B1(t) =2t (1− t) and B2(t) = t2, for i ∈ {0, 1, 2}, and wi are weight associated to thecontrol points Pi. For a standard RQBC w0 and w2 are equal to 1, while w1 canbe used to control the type the conic defined by the curve.Rational Biquadratic Bezier Surfaces (RBBS) are defined by a tensor productof two RQBC by:
−−−−−−−→OM (u, v) =
12∑
i=0
2∑
j=0
wijBi (u)Bj (v)
2∑
i=0
2∑
j=0
wijBi (u)Bj (v)−−−→OPij (2)
More details on Bezier curves and surfaces can be found in [5, 9].
3 Modeling Circular Arcs Using RQBC
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. Inthis section, we give some results on the expression of circular arcs using RQBC.
Theorem 1 shows how to define a circle from two points and two tangents onthese points. Theorem 2 presents how to compute the middle control point ofthe RQBC that represent a given circular arc. Theorem 3 shows how to computethe weight of the middle the middle control point of the RQBC that represent agiven circular arc. Figure 1 shows the modeling of circular arcs using RQBC.
The geometric construction used for theorems 1, 2 and 3 is as follows: C(O0, R)is a circle of center O0 and radius R. Segments [P0P1] and [P2P1] are tangentsto the circle at points P0 and P2. I1 is the midpoint of segment [P0P2]. P themedian plane of segment [P0P2].
Theorem 1. Circle from two points and tangents at these points
– Circle C(O0, R) exists if and only if P1 ∈ P and P1 /∈ [P0P2]. The radiusR = O0P0 and the center O0 is given by formula:
−−−→P1O0 = t0
−−→P1I1 t0 =
P0P21−−→
I1P1 • −−−→P0P1
(3)
Fig. 1. Modeling circular arcs by RQBC
– In the plane determined by C, the geometric angle P0O0P2 is less than π.This means that if we take the parameterization γ of the circle in terms ofcosine and sine such as P0 = γ (θ0), P2 = γ (θ1), we have |θ0 − θ1| < π.
Theorem 2. Computing control point P1 when the center of the circle is knownThe RQBC is the arc of the circle C passing through P0 and P2. In this case, thecontrol point P1 verifies:
−−→I1P1 = t1
−−→O0I1 t1 =
−−−→O0P0 • −−→I1P0−−−→O0P0 • −−→O0I1
(4)
Theorem 3. Computing the weight w1.The RQBC defined by control points P0, P1 and P2 and the weight w1 is acircular arc if and only if the following condition hold:
|1 + w1| = |O0I1 + w1O0P1| (5)
The RQBC defines the small arc of circle if:
w1 =O0I1 −R
R−O0P1=
O0I1 −O0P0
O0P0 −O0P1> 0 (6)
It defines the big arc of circle if:
w1 = −O0I1 + R
R + O0P1= −O0I1 + O0P0
O0P0 + O0P1< 0 (7)
Proofs af these theorems can be easily obtained by combining properties ofRQBC with those of the circle and scalar product.
4 Barycentric Properties of RBBSs
Let S0 be the Rational Biquadric Bezier Surface (RBBS) defined according toformula (2) by control points (Pij)0≤i,j≤2 and weights (wij)0≤i,j≤2 with w00 =
w02 = w20 = w22 = 1. In order to represent surfaces with spherical curvaturesby surface S0, we should have the following constraints on control points: P01
belongs to the median plane of [P00P02], P10 belongs to the median plane of[P00P20], P21 belongs to the median plane of [P20P22] and P12 belongs to themedian plane of [P02P22].The following theorem introduces a series of interesting barycentric propertiesof RBBSs that helps in the expression of quadrics as RBBSs
Theorem 4. Barycentric properties of RBBSs
1. Let I0, J0, I2 and J2 be respectively the midpoints of the segments [P00P02],[P00P20], [P20P22] and [P02P22]. We have the following four relations:
−−−−−−−−→OM
(0,
12
)=
11 + w01
(−−→OI0 + w01
−−−→OP01
)(8)
−−−−−−−−→OM
(1,
12
)=
11 + w21
(−−→OI2 + w21
−−−→OP21
)(9)
−−−−−−−−→OM
(12, 0
)=
11 + w10
(−−→OJ0 + w10
−−−→OP10
)(10)
−−−−−−−−→OM
(12, 1
)=
11 + w12
(−−→OJ2 + w12
−−−→OP12
)(11)
2. Let G0 be the isobarycenter of points P00, P02, P20, P22 and G2 the barycenterof weighted points (P10, w10), (P01, w01), (P12, w12), (P21, w21). We define thevalue w = w01 + w10 + w12 + w21 and G1 the barycenter of weighted points(G0, 2) and (G2, w).The point M
(12 , 1
2
)verifies the two following formulas:
−−−−−−−−→OM
(12,12
)=
12 + w + 2w11
((2 + w)
−−→OG1 + 2w11
−−−→OP11
)(12)
w11
−−−−−−−−−−→M
(12,12
)P11 = −2 + w
2
−−−−−−−−−−→M
(12,12
)G1 (13)
>From the latest formula we deduce that P11 belongs to the line(M
(12 , 1
2
)G1
).
3. Let G3 be the barycenter of weighted points (P00, 9), (P20, 9), (P02, 1), (P22, 1),(P01, 6w01), (P21, 6w21), (P10, 18w10), (P12, 2w12), and W1 = 20 + 6w01 +18w10 + 2w12 + 6w21.The point M
(12 , 1
4
)verifies the two following formulas:
−−−−−−−−→OM
(12,14
)=
1W1 + 12w11
(W1−−→OG3 + 12w11
−−−→OP11
)(14)
(W1 + 12w11)−−−−−−−−−→G3M
(12,14
)= 12w11
−−−−→G3P11 (15)
>From the latest formula we deduce that P11 belongs to the line(G3M
(12 , 1
4
)).
4. Let G4 be the barycenter of weighted points (P00, 9), (P20, 1), (P02, 9), (P22, 1),(P10, 6w10), (P12, 6w12), (P01, 18w01), (P21, 2w21), and W2 = 20 + 6w10 +18w01 + 2w21 + 6w12.The point M
(14 , 1
2
)verifies the two following formulas:
−−−−−−−−→OM
(14,12
)=
1W2 + 12w11
(W2−−→OG4 + 12w11
−−−→OP11
)(16)
(W2 + 12w11)−−−−−−−−−→G4M
(14,12
)= 12w11
−−−−→G4P11 (17)
>From the latest formula we deduce that P11 belongs to the line(G4M
(14 , 1
2
)).
Proof.
1. In order to prove the expressions (8) and (9) for points−−−−−−−−→OM
(0,
12
)and
−−−−−−−−→OM
(1,
12
), let us recall that B0 (0) = 1, B1 (0) = B2 (0) = 0, B2 (1) = 1,
B1 (1) = B0 (1) = 0, B1
(12
)= 1
2 , B0
(12
)= B2
(12
)= 1
4 , and if I is themidpoint of segment [AB], then for every point O we have
−→OA+
−−→OB = 2
−→OI.
By Formula (2), The point−−−−−−−−→OM
(0,
12
)on the RBBS is:
−−−−−−−−→OM
(0,
12
)=
12∑
i=0
2∑
j=0
wijBi (0)Bj
(12
)2∑
i=0
2∑
j=0
wijBi (0) Bj
(12
)−−−→OPij
=1
2∑
j=0
w0jBj
(12
)2∑
j=0
w0jBj
(12
)−−−→OP0j
=1
w00
4+
w01
2+
w02
4
(w00
4−−−→OP00 +
w01
2−−−→OP01 +
w02
4−−−→OP02
)
=1
14 + w01
2 + 14
(14
(−−−→OP00 +
−−−→OP02
)+
w01
2−−−→OP01
)
=1
12 + w01
2
(142−−→OI0 +
w01
2−−−→OP01
)=
21 + w01
(12−−→OI0 +
12w01
−−−→OP01
)
=1
1 + w01
(−−→OI0 + w01
−−−→OP01
)
On the other hand, point−−−−−−−−→OM
(1,
12
)on the RBBS is:
−−−−−−−−→OM
(1,
12
)=
12∑
i=0
2∑
j=0
wijBi (1)Bj
(12
)2∑
i=0
2∑
j=0
wijBi (1) Bj
(12
)−−−→OPij
=1
2∑
j=0
w2jBj
(12
)2∑
j=0
w2jBj
(12
)−−−→OP2j
=1
w20
4+
w21
2+
w22
4
(w20
4−−−→OP20 +
w21
2−−−→OP21 +
w22
4−−−→OP22
)
=1
14 + w21
2 + 14
(14
(−−−→OP20 +
−−−→OP22
)+
w21
2−−−→OP21
)
=1
12 + w21
2
(142−−→OI2 +
w21
2−−−→OP21
)=
21 + w21
(12−−→OI2 +
w21
2−−−→OP21
)
=1
1 + w21
(−−→OI2 + w21
−−−→OP21
)
Expressions (10) and (11) of−−−−−−−−→OM
(12, 0
)and
−−−−−−−−→OM
(12, 1
)can be prooved
in a similar way.
2. Proof of the expression (12) for point−−−−−−−−→OM
(12,12
):
−−−−−−−−→OM
(12,12
)=
12∑
i=0
2∑
j=0
wijBi
(12
)Bj
(12
)2∑
i=0
2∑
j=0
wijBi
(12
)Bj
(12
)−−−→OPij
First, let us consider the denominator of this fraction:2∑
i=0
2∑
j=0
wijBi
(12
)Bj
(12
)=
2∑
i=0
Bi
(12
) (wi0
4+
wi1
2+
wi2
4
)
=
w00
4+
w01
2+
w02
44
+
w10
4+
w11
2+
w12
42
+
w20
4+
w21
2+
w22
44
=
w00 + w02 + w20 + w22
4+
w01 + w10 + w21 + w12
24
+
w11
22
=w00 + w02 + w20 + w22
16+
w01 + w10 + w21 + w12
8+
w11
4
=2 + w + 2w11
8
Second, in the same way the nominator can be expressed as:
2∑
i=0
Bi
(12
) (wi0
4−−−→OPi0 +
wi1
2−−−→OPi1 +
wi2
4−−−→OPi2
)
=w00
16−−−→OP00 +
w01
8−−−→OP01 +
w02
16−−−→OP02 +
w10
8−−−→OP10 +
w11
4−−−→OP11 +
w12
8−−−→OP12+
wi0
16−−−→OP20 +
w21
8−−−→OP21 +
w22
16−−−→OP22
=(
116−−−→OP00 +
116−−−→OP02 +
116−−−→OP20 +
116−−−→OP22
)+
(w01
8−−−→OP01 +
w10
8−−−→OP10 +
w12
8−−−→OP12 +
w21
8−−−→OP21
)+
w11
4−−−→OP11
=14−−→OG0 +
w
8−−→OG2 +
w11
4−−−→OP11
So we have:−−−−−−−−→OM
(12,12
)=
12+w+2w11
8
(14−−→OG0 +
w
8−−→OG2 +
w11
4−−−→OP11
)
=8
2 + w + 2w11
(14−−→OG0 +
w
8−−→OG2 +
w11
4−−−→OP11
)
=1
2 + w + 2w11
(2−−→OG0 + w
−−→OG2 + 2w11
−−−→OP11
)
=1
2 + w + 2w11
((2 + w)
−−→OG1 + 2w11
−−−→OP11
)
>From these results, the proof of expression (13) is straightforward:
(2 + w + 2w11)−−−−−−−−−−−−−−−−→M
(12,12
)M
(12,12
)=(2 + w)
−−−−−−−−−−→M
(12,12
)G1+
2w11
−−−−−−−−−−→M
(12,12
)P11
So we have:
w11
−−−−−−−−−−→M
(12,12
)P11 = −2 + w
2
−−−−−−−−−−→M
(12,12
)G1
3. Before starting the proof of expression (14) for point−−−−−−−−→OM
(12,14
), recall
that B0
(14
)= 9
16 , B1
(14
)= 3
8 et B2
(14
)= 1
16 . By formula (2) we have:
−−−−−−−−→OM
(12,14
)=
12∑
i=0
2∑
j=0
wijBi
(12
)Bj
(14
)2∑
i=0
2∑
j=0
wijBi
(12
)Bj
(14
)−−−→OPij
The denominator can be easily reduced as:
2∑
i=0
2∑
j=0
wijBi
(12
)Bj
(14
)=
2∑
i=0
Bi
(12
)(9wi0
16+
6wi1
16+
wi2
16
)
=9w00 + 9w20 + w02 + w22 + 6w01 + 6w21 + 18w10 + w12 + 12w11
64
=20 + 6w01 + 6w21 + 18w10 + w12 + 12w11
64=
W1 + 12w11
64
using this result and developing the nominator lead to the intended proof:−−−−−−−−→OM
(12,14
)=
64W1 + 12w11
(964−−−→OP00 +
6w01
64−−−→OP01 +
w02
64−−−→OP02 +
1864−−−→OP10 +
12w11
64−−−→OP11
)+
64W1 + 12w11
(2w02
64−−−→OP12 +
964−−−→OP20 +
6w21
64−−−→OP21 +
164−−−→OP22+
)
=1
W1 + 12w11
(W1−−→OG3 + 12w11
−−−→OP11
)
>From these results we can deduce that : (W1 + 12w11)−−−−−−−−−→G3M
(12,14
)=
12w11−−−−→G3P11 which is the expression (15)
4. The proof of expressions (16) and (17) 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 revolutionquadric into a rational biquadric Bezier patch. Some conversion illustrations aregiven. Terms quadric patch, sphere patch, cone patch or cylinder patch will beused to refer to the part of the quadric we are converting. A set of conversionillustrations is given in section 6. Following is the conversion algorithm:
Given: A quadric surface defined by a parametric map Γ , and a patch on thisquadric delimited by parameter values: θ0, θ1, ψ0 and ψ1 with |θ0 − θ1| < 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, 1] defined by ninecontrol points Pij and nine weights wij , 0 6 i, j < 2. Weights wij will be equalto one except w10, w01, w21 et w12 that can have positif or negative values.
Proceeding:
1. Obtain corner control points directly by: P00 = Γ (θ0, ϕ0), P02 = Γ (θ1, ϕ0),P20 = Γ (θ0, ϕ1), P22 = Γ (θ1, ϕ1). and |ϕ0 − ϕ1| < π .
2. Find centers of the four circles of curvature using equation (3) of theorem 1.3. Find control points P01, P10, P12 and P21 on the median planes of curvature
lines using equation (4) of theorem 2.4. Calculate weights w10, w01, w21 and w12 using equations (6) and (7) of
theorem 3.5. Consider two RQBC γu and γv defined on the borders of the region to be
converted. Control points and the weight of γu are P00, P10, P20 and w10.Those of γv are P00, P01, P02 and w01.– Find θ2, solution of equation γu
(12
)= Γ (θ2, ψ0).
– Find ψ2, solution of equation γv
(12
)= Γ (θ0, ψ2).
– Find ψ3, solution of equation γv
(14
)= Γ (θ0, ψ3).
6. Compute the last control point P11 as the intersection of lines (G1Γ (θ2, ψ2))and (G3Γ (θ2, ψ3)). the construction of G1 and G3 are given in theorem 4.
7. Compute weight w11, using equation (13) of theorem 4
(a) (b) (c) (d)
Fig. 2. 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 itrepresents exactly the same region as the converted patch. (c) The second patch, it isdelimited by the north pole of the sphere. (d) The obtained equivalent RBBS.
6 Results on Quadric Patches Conversion
6.1 Conversion to Standard RBBSs
Cases presented in this sub-section concern only the conversion of quadric patchesinto standard RBBSs where all weight associated to control points are positif.The conversion algorithm of section 5 is applied to determine control points aswell as weights of the RBBS that represents the quadric patch.
Conversion of Sphere Patches: Given a sphere S defined by the parametricmap Γ as:
Γ : [0; 2π]×[−π
2;π
2
]−→ E
(θ ψ) 7−→ Γ (θ ψ) = (R cos (θ) cos (ψ) , R sin (θ) cos (ψ) , R sin (ψ))(18)
One can notice here that θ2 and ψ2 (values needed to define lines containing
control point P11 are given by: θ2 = arctan(
γuy ( 12 )
γux( 12 )
)and ψ2 = arcsin
(γuz ( 1
2 )R
).
Figure 2 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 aspecial consideration. For the north pole, we have ψ1 = π
2 and P02 = P12 = P22.The RQBC defined by P02, P12 and P22 is a point, but its nature should still thesame as the RQBC defined by P00, P10 and P20, so we must have w12 = w10.
(a) (b) (c) (d)
Fig. 3. A cone patch (a) and a cylinder patch (c) with their equivalent RBBS repre-sentations (b) and (d) respectively
Conversion of Cone or Cylinder Patches: Figure 3 shows a part of a coneand a part of a cylinder with their equivalent RBBS representations. We establishher that, in the case of the cone, it is possible to take the vertex of the cone asa delimiter of the part to be converted. It is a similar situation to the pole of asphere and we have: P02 = P12 = P22 and w12 = w01.
(a) (b) (c)
Fig. 4. Conversion of a part of a cylinder (a) into two RBBSs: (b) is a standard RBBS,(c) is a RBBS with negative weights.
6.2 Conversion to RBBSs with Negative Weights
By allowing negative weights for our RBBSs, we can see that it is also possibleto convert more complex patches (e.g. the whole cone or the whole cylinder).Figures 4, 5 show respectively conversions of larger patches of a cylinder and acone into two RBBSs. For each case, the same nine control points and twelveweights define both of the obtained RBBSs. Positive weights define the smallpatch (subfigures b), while negative weights allows to define the large patch(subfigures c). In the two (a) subfigures, the obtained RBBSs are positioned onthe 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 barycen-tric properties of rational biquadratic Bezier surfaces, thanks to the symmetryproperty of circles and Bernstein polynomials. A quadrics to RBBSs conversionalgorithm based on these barycentric properties is also proposed. Several conver-sion experimental cases are given to show the results obtained by our conversionalgorithm.
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