Chapter 13

Quadric Surfaces

The quadrics are all surfaces that can be expressed as a second degree polynomialin x, y and z. They include important principle shapes such as those shown inFigure 13.1.

The general implicit form for a 3D quadric surface can be written in homoge-neous x, y, z, w coordinates as:

ax2 + 2bxy+ 2cxz+ 2dxw+ ey2 + 2fyz+ 2gyw+hz2 + 2izw+ jw2 = 0. (13.1)

Setting w = 1, this provides the ability to position the quadrics in space. Thisequation can also be written in the matrix form:

xtQx = 0, (13.2)

where x =

xyzw

, xt =[x y z w

], and Q =

a b c db e f gc f h id g i j

(multiply this out and you will see that you get the original polynomial form.You will also see why we put the 2s in the original equation).

Now, if we wish to raytrace a quadric, we insert the ray equation x = p + tuinto Equation 13.2, giving

(p + tu)tQ(p + tu) = 0,

orptQp + tptQu + tutQp + t2utQu = 0.

Since Q is symmetric, the two middle terms of this equation are identical, so wecan rearrange and write

t2utQu + 2tutQp + ptQp = 0. (13.3)

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82 CHAPTER 13. QUADRIC SURFACES

sphere

x2 + y2 + z2 − 1 = 0

ellipsoid

ax2 + by2 + cz2 − 1 = 0cylinder

x2+z2−1 = 0

cone

x2 + z2 − y2 = 0

hyperbolic paraboloid(saddle)

x2 − z2 − y = 0

paraboloid

x2+z2−y = 0

hyperboloid of onesheet

x2 + z2 − y2 − 1 = 0

hyperboloid of twosheets

x2 + z2 − y2 + 1 = 0

Figure 13.1: The important quadric surfaces.

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Note that Equation 13.3 is quadratic in t, so by the quadratic formula it hassolutions

t =−utQp±

√(utQp)2 − (utQu)(ptQp)

utQu. (13.4)

This gives a general form for all of the quadrics that is completely and uniquelyspecified by the matrix Q. This representation is especially nice, since we cantransform the quadric into any desired coordinate system via the transformation

Q′ = M−1Q(M−1)t,

where M is the transform from the object coordinates in which the quadric isdefined to the desired scene coordinates.

Because of this, we need only consider the important canonical principle quadricshapes, like those in Figure 13.1, and then apply the appropriate transform tocreate the shape desired, before transforming to world coordinates. For example,a sphere of radius 1, centered at the origin, is given by

Qsphere =

1 0 0 00 1 0 00 0 1 00 0 0 −1

.To make a sphere of radius 3, we simply scale via

M =

3 0 0 00 3 0 00 0 3 00 0 0 1

, with M−1 =

1/3 0 0 00 1/3 0 00 0 1/3 00 0 0 1

,so that

M−1Qsphere(M−1)t =

1/3 0 0 00 1/3 0 00 0 1/3 00 0 0 1

1 0 0 00 1 0 00 0 1 00 0 0 −1

1/3 0 0 00 1/3 0 00 0 1/3 00 0 0 1

=

1/9 0 0 00 1/9 0 00 0 1/9 00 0 0 −1

,which gives a sphere of radius 3 =

√9.

We can go further. For example, we can make an ellipsoid with desired radius

84 CHAPTER 13. QUADRIC SURFACES

in the x, y and z directions by a non-uniform scale

M =

a 0 0 00 b 0 00 0 c 00 0 0 1

.Similarly, we can translate, rotate or shear these shapes to make a large varietyof shapes from this one representation.

Raytracing is as simple as solving Equation 13.4. Well – not quite! Except forthe sphere and ellipsoid, all of the quadrics need caps to be useful, since theyare open, infinite surfaces.

y=1

y=0

For example, the cylinder x2 + z2 − 1 = 0 is infinitein the y direction. We can apply caps at y = 0 andy = 1 (for example) by intersecting the cylinder withthe planes y = 0 and y = 1.

When we shoot a ray, we look for all intersectionsbetween the ray and the cylinder and the two planes,and sort these in order by distance along the ray.From this we can tell if the ray hits the truncatedcylinder (between y = 0 and y = 1), and if it hits it,if the hit is on one of the caps or on the body.

Let c1, c2 be the cap plane hits, and let b1, b2 behits on the cylinder body. The value of each is theray parameter at the hit, and assume (without lossof generality) that c1 < c2 and b1 < b2. We haveseveral non degenerate cases and some special cases.The general cases are:

1. c1, c2, b1, b2 : miss

2. c1, b1, c2, b2 : hit body, exit cap

3. c1, b1, b2, c2 : hit body, exit body

4. b1, c1, b2, c2 : hit cap, exit body

5. b1, c1, c2, b2 : hit cap, exit cap

6. b1, b2, c1, c2 : miss

The special cases are when one of the equations has only one solution, forexample if the ray is tangent to the cylinder, or no solutions, like when the raymisses the cylinder, or is parallel to the caps.

Note that it is also possible to find a parameterization for each of the quadrics.

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In the last chapter, we already discussed how to do this for a sphere usingspherical coordinates. Parameterizing a cylinder or a cone is straightforwardusing cylindrical coordinates. In this case, the angle parameter measures aroundthe body of the cylinder or cone, and the second parameter measures height. Ifa uniform spacing is desired in the vertical direction all of the other quadricsexcept saddle can also be parameterized using cylindrical coordinates.