1. Creating Meshes Through Parameterized Functions
Lecture 26
Mon, Oct 29, 2007

2. Surfaces Defined by Functions
Let F(x, y, z) = 0 be an equation that implicitly defines a 2-dimensional surface in 3-dimensional space.
For example,
x2 + y2 + z2 – 1 = 0 defines a sphere.
x2 + y2 – z2 = 0 defines a cone.

3. Parameterizing the Surface
Often the surface can be parameterized in terms of two parameters s and t, based on some intrinsic properties of the surface.
Let x, y, and z be given in terms of s and t, with a  s  b, c  t  d.
x = x(s, t)
y = y(s, t)
z = z(s, t)
Often [a, b] and [c, d] are [0, 1] or [0, 2].

4. Parameterizing the Surface
Then P(s, t) = (x, y, z) is a point on the surface.

5. Example: A Sphere of Radius 1
Use spherical coordinates, where
s is the angle around the vertical axis (longitude),
t is the angle up from the equator (latitude).
Then let
x = cos t cos s,
y = cos t sin s.
z = sin t,
with 0  s  2 and -/2  t  /2.

6. Example: A Sphere of Radius 1
Then P(s, t) = (cos t cos s, cos t sin s, sin t) is a point on the surface.
Check:
x2 + y2 + z2 – 1 = cos2 t cos2 s + cos2 t sin2 s
+ sin2 t – 1,
= cos2 t + sin2 t – 1,
= 0.

7. A Spherical Mesh
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8. Example: A Cone of Radius 1 and Height 1
To define the surface of a cone, let
x = t cos s
y = t sin s
z = t
with 0  s  2 and 0  t  1.
s represents an angle around the central axis.
t represents the height and the radius of a horizontal cross-section

9. Example: A Cone of Radius 1 and Height 1
Then
s represents an angle around the central axis.
t represents the height and the radius of a horizontal cross-section
Then P(s, t) = (t cos s, t sin s, t) is a point on the surface of the cone.
Check:
x2 + y2 – z2 = t2 cos2 s + t2 sin2 s – t2 = 0.

10. Contours
An s-contour is the 1-dimensional curve we get if we hold t fixed and let s vary.
A t-contour is the 1-dimensional curve we get if we hold s fixed and let t vary.
Typically, we get a different s-contour for each value of t and a different t-contour for each value of s.

11. Example: Contours on a Cone
Let t = ½.
Then x = ½ cos s, y = ½ sin s, z = ½.
The s-contour is the curve
x2 + y2 = ¼
which is a circle of radius ½.
This is a slice of the cone parallel to the base.

12. Example: Contours of a Cone

13. Example: Contours of a Cone
Let s = /2.
Then x = 0, y = t, z = t.
The t-contour is the curve
y = z
which is a straight line.
This is a slice of the cone perpendicular to the base.

14. Example: Contours of a Cone

15. A Conical Mesh
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16. Finding Normal Vectors
At a point P(s, t),
The tangent to the s-contour is given by
P/s.
The tangent to the t-contour is given by
P/t.
Thus, the unit normal to the surface is
n = (P/s)  (P/t),
normalized.

17. Example: Finding Normals
Find the normals to the surface of the cone defined by x2 + y2 = z2.
Let P(s, t) = (t cos s, t sin s, t).
P/s = (-t sin s, t cos s, 0).
P/t = (cos s, sin s, 1).
N = (t cos s, t sin s, -t).
Normalize to n = (cos s, sin s, -1)/2.

18. The Direction of the Normals
Caution: This proceed is as likely to produce vectors that point “inward” as it is to produce vectors that point “outward.”
If the vectors point inward, then attach a negative sign to each component to reverse the direction.

19. The Toroidal Mesh
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20. The Cylindrical Mesh
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21. Several Meshes in a Scene
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22. A Snowman
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23. A Lamp and Lampshade

24. The Mesh Class
mesh.h
mesh.cpp