lecture 25 robb t. koether - hampden-sydney...

Parameterized MeshesLecture 25

Robb T. Koether

Hampden-Sydney College

Mon, Oct 26, 2015

Robb T. Koether (Hampden-Sydney College) Parameterized Meshes Mon, Oct 26, 2015 1 / 30

Outline

1 Parameterized SurfacesA Parameterized SphereA Parameterized Paraboloid

2 ContoursContours of a SphereContours of a Paraboloid

3 Calculating Normal Vectors

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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.


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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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 .


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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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.


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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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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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.



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 .



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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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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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.


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).


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).


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).


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.


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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Assignment 14

Assignment 14

Find a parameterization of the cone x2 + z2 = y2, including thenormal vectors.


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Assignment

AssignmentRead pp. 359 - 368, Classic Lighting Model.


lecture 25 robb t. koether - hampden-sydney...

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