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Parameterized MeshesLecture 25
Robb T. Koether
Hampden-Sydney College
Mon, Oct 26, 2015
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Outline
1 Parameterized SurfacesA Parameterized SphereA Parameterized Paraboloid
2 ContoursContours of a SphereContours of a Paraboloid
3 Calculating Normal Vectors
4 Assignment 14
5 Assignment
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Outline
1 Parameterized SurfacesA Parameterized SphereA Parameterized Paraboloid
2 ContoursContours of a SphereContours of a Paraboloid
3 Calculating Normal Vectors
4 Assignment 14
5 Assignment
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Surfaces Defined by Functions
Let F (x , y , z) = 0 be an equation that implicitly defines a2-dimensional surface in 3-dimensional space.For example,
x2 + y2 + z2 − 1 = 0 defines a sphere.x2 + z2 − y2 = 0 defines a cone.
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Parameterizing 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)
Typically, [a,b] and [c,d ] are [0,1] or [0,2π].Then P(s, t) = (x , y , z) is a point on the surface.
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Outline
1 Parameterized SurfacesA Parameterized SphereA Parameterized Paraboloid
2 ContoursContours of a SphereContours of a Paraboloid
3 Calculating Normal Vectors
4 Assignment 14
5 Assignment
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A Sphere of Radius 1
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 sin sy = sin tz = cos t cos s,
with 0 ≤ s ≤ 2π and −π2 ≤ t ≤ π
2 .
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A Sphere of Radius 1
Example (A Sphere of Radius 1)Then
P(s, t) = (cos t sin s, sin t , cos t cos s)
is a point on the surface.Check:
x2 + y2 + z2 − 1 = (cos t sin s)2 + (sin t)2 + (cos t cos s)2 − 1
= cos2 t sin2 s + sin2 t + cos2 t cos2 s − 1
= cos2 t(
sin2 s + cos2 s)
+ sin2 t
= cos2 t + sin2 t − 1= 1− 1= 1.
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Outline
1 Parameterized SurfacesA Parameterized SphereA Parameterized Paraboloid
2 ContoursContours of a SphereContours of a Paraboloid
3 Calculating Normal Vectors
4 Assignment 14
5 Assignment
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A Paraboloid
Example (A Paraboloid)s represents an angle around the central axis.t represents the radius of a circular horizontal cross-section.The paraboloid of height 1 is defined by
x = t sin s
y = t2
z = t cos s
with 0 ≤ s ≤ 2π and 0 ≤ t ≤ 1.
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A Paraboloid
Example (A Paraboloid)Then
P(s, t) = (t sin s, t2, t cos s)
is a point on the surface of the paraboloid.Check:
x2 + z2 − y = t2 sin2 s + t2 cos2 s − t2
= t2(
sin2 s + cos2 s)− t2
= t2 − t2
= 0.
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Outline
1 Parameterized SurfacesA Parameterized SphereA Parameterized Paraboloid
2 ContoursContours of a SphereContours of a Paraboloid
3 Calculating Normal Vectors
4 Assignment 14
5 Assignment
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Contours
Definition (s- and t-Contours)An s-contour is the 1-dimensional curve we get if we hold t fixed andlet s vary over its domain. A t-contour is the 1-dimensional curve weget if we hold s fixed and let t vary over its domain.
Typically, we get a different s-contour for each value of t and adifferent t-contour for each value of s.
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Outline
1 Parameterized SurfacesA Parameterized SphereA Parameterized Paraboloid
2 ContoursContours of a SphereContours of a Paraboloid
3 Calculating Normal Vectors
4 Assignment 14
5 Assignment
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Contours on a Sphere
Example (Contours on a Sphere)Recall that the sphere is defined by
x = cos t sin sy = sin tz = cos t cos s.
Let t = π3 and find the s-contour.
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Contours on a Sphere
Example (Contours on a Sphere)We get
x = cosπ
3sin s =
12
sin s
y = sinπ
3=
√3
2
z = cosπ
3cos s =
12
cos s.
This is the circlex2 + z2 =
14
at the height y =√
32 .
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Contours on a Sphere
Example (Contours on a Sphere)Now let s = π
3 and find the t-contour.We get
x = cos t sinπ
3=
√3
2cos t
y = sin t
z = cos t cosπ
3=
12
cos t .
This is a circle of radius 1 in the plane x =√
3z.
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Outline
1 Parameterized SurfacesA Parameterized SphereA Parameterized Paraboloid
2 ContoursContours of a SphereContours of a Paraboloid
3 Calculating Normal Vectors
4 Assignment 14
5 Assignment
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Contours on a Paraboloid
Example (Contours on a Paraboloid)
Find the s-contour of the paraboloid for s = π3 and t = 1
2 .For the paraboloid,
Find the s-contour when t = 12 .
Find the t-contours when s = 0 and when s = π2 .
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Outline
1 Parameterized SurfacesA Parameterized SphereA Parameterized Paraboloid
2 ContoursContours of a SphereContours of a Paraboloid
3 Calculating Normal Vectors
4 Assignment 14
5 Assignment
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Finding Normal Vectors
At a point P(s, t),A vector tangent to the s-contour is given by
∂P∂s
.
A vector tangent to the t-contour is given by
∂P∂t.
Thus, a normal vector to the surface is
n =∂P∂s× ∂P
∂t.
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Finding Normals
Example (Finding Normals)Find the normals to the surface of the sphere defined byx2 + y2 + z2 = 1.Let
P(s, t) = (cos t sin s, sin t , cos t cos s).
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Finding Normals
Example (Finding Normals)Then
∂P∂s
= (cos t cos s,0,− cos t sin s)
∂P∂t
= (− sin t sin s, cos t ,− sin t cos s).
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Finding Normals
Example (Finding Normals)Then
n = (cos2 t sin s, cos t sin t cos2 s + cos t sin t sin2 s, cos2 t cos s)
= (cos2 t sin s, cos t sin t + cos t sin t , cos2 t cos s)
= cos t(cos t sin s, sin t , cos t cos s)
= cos t(x , y , z)
N =n|n|
= (x , y , z).
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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.”It depends on the parametrization.If the vectors point inward, then use the negative of the vector.
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The Paraboloid Mesh
Example (The Paraboloid Mesh)Find the normal vectors for a paraboloid.Which way do they point?Which way should they point?
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Outline
1 Parameterized SurfacesA Parameterized SphereA Parameterized Paraboloid
2 ContoursContours of a SphereContours of a Paraboloid
3 Calculating Normal Vectors
4 Assignment 14
5 Assignment
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Assignment 14
Assignment 14
Find a parameterization of the cone x2 + z2 = y2, including thenormal vectors.
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Outline
1 Parameterized SurfacesA Parameterized SphereA Parameterized Paraboloid
2 ContoursContours of a SphereContours of a Paraboloid
3 Calculating Normal Vectors
4 Assignment 14
5 Assignment
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Assignment
AssignmentRead pp. 359 - 368, Classic Lighting Model.
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