linear interpolation

Linear Interpolation

• If a=1,

• If a=0,

• If a=0.5,

Midterm_Grade=a*Old_Grade+(1-a)*New_Grade

The original grade

The grade after you redo Some problems

Midterm_Grade = 1*Old_Grade+0*New_Grade = Old_Grade

Midterm_Grade = 0*Old_Grade+1*New_Grade = New_Grade

Midterm_Grade = 0.5*Old_Grade+0.5*New_Grade =(Old_Grade+New_Grade)*0.5


• Now if you change the grade into two vertices:

V0

V1

Va=a*V1+(1-a)*V0

a=1

a=0

V0.5

a=0.5a=0.25

a=0.75

a>1

a<0This is how we define a line, a line segment, or a ray.


• Now if know V1, V2, and V, how do you know a instead?

• We know:

V0

V1

V=a*V1+(1-a)*V0

Va

V-V0=a*(V1-V0)

a=|V-V0|/|V1-V0|

|V1-V0|

|V-V0|

Linear Interpolation• Now if you have two values C0 and C1 (color, depth, texture

coordinates…) at V0 and V1, how to get the value Ca at V:

V0

V1

Ca=a*C1+(1-a)*C0

Va

|V1-V0|

|V-V0|

C0

C1

Ca???

a=|V-V0|/|V1-V0|

Midterm Grading Formula

• Option 1: No new grade, just curve it. (a=1)

• Option 2: Redo two questions, take the average and curve it. (a=0.5)

• Option 3: Redo two questions, take the average but no curve. (a=0.5, b=1)

• Option 4: Redo all questions, take the average but no curve. (a=0.5, b=1)

Midterm_Grade=(a*Old_Grade+(1-a)*New_Grade)b

a: Interpolation coefficient b: Curve coefficient

Midterm Grading Formula• Redo two questions• Pro: less work for you (and for me)• Con: depends on the choice of questions, not very

fair for some people:• A: 3+3+3+3+3=15 (old); 3+5+5+3+3=19 (new).• B: 5+0+0+5+5=15 (old); 5+5+5+5+5=25 (new).

• Redo all questions• Con: more work• Pro: you get the second chance to learn; and it is

relatively fair

Midterm Grading Formula

• Details:

• Return the graded old exam package to you now (on Wednesday). So you know which answer is wrong and which is right.

• Only the old grade, but not the exam package. You know your overall performance, but don’t know which answer is wrong exactly.

• After you submit your new answers, you will get the graded old exam package.

Homogenous Space Vs. 3D Space

xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

A point in the 3D Space

xyz1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

A point in the homogenous Space (4D)

xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥


xyz1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

A point in the homogenous Space

Homogenous Space Vs. 3D Space

a wb wc w

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥


abcw

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

A point in the homogenous Space

a wb wc ww w

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥=

a wb wc w1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

What if w is not 1? Scale it by 1/w!

Projection and Transformation

xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥


xyz1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

A point in the homogenous Space (4D)

a b c de f g hi j k lm n o p

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

xyz1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

Multiply it by your matrix

abcw

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

Your result in the homogenous Space (4D)

a wb wc w

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

Your solution in 3D!

Projection and Transformation

xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

abc1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

Transformation won’t change w. w is always 1.

xyz1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

M

xyz1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

abcw

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

Projection changes w!

xyz1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

P

xyz1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

a wb wc w

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

Depth Value(Z value)

2D image location

Transformation

x 'y 'z '1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

A point in an absolute space

xyz1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

M

xyz1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

Its changed position in the absolute space

One way to understand the transformation:

Transformation

x 'y 'z '1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

A point in one space

xyz1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

M

xyz1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

The same point, but in a new space

Another way to understand the transformation:

Transformationx 'y 'z '1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

A point in the local space(or called the model space)

xyz1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

Mmodelview

xyz1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

A point in the eye space(or called the camera space)

x 'y 'z '1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

A point in the local space(or called the model space)

xyz1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

Mmodelview−1

x'y'z'1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

A point in the eye space(or called the camera space)

Transformation

glLoadIdentity();glRotatef(…);glTranslatef(…);glScalef(…);glBegin(GL_POINTS);glVertex3fv(v);glEnd();

I

R

T

S

Mmodelview =I

Mmodelview =IR

Mmodelview =IRT

Mmodelview =IRTS

(eye space -> eye space)

(space No. 1 -> eye space)



Mmodelviewv=IRTSv=v'(space No. 3 -> eye space)

Transformation

Mmodelview =

m 00 m 01 m 02 m 03

m 10 m 11 m 12 m 13

m 20 m 21 m 22 m 23

1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

The matrix not only transforms a vertex from the localspace into the eye space, it also tells us how the local space looks like in the eye space:

The local OriginThe local Y axis

The local X axis The local Z axis

Transformation

glLoadIdentity();//Camera motions (also as transformation)

//Do more transformations

glBegin(GL_POINTS);glVertex3fv(v);glEnd();

Eye space

World space

Local space

Mviewing

Mmodeling

Mmodelview

OpenGL only has the modelview matrix, which really contains two steps: viewing and modeling.

Shading

V0 V1

Phong ShadingVs

f(Vs)f(V0) f(V1)

• Now you did projection, you have polygons in 2D.• You do rasterization, so you have scanlines. Each line have

two endpoints and you have a lot of pixels between them.• Suppose you have a function f(x) that can give you a value at

any point, how do you use it to get a value for each pixel?

Shading

V0 V1




Flat ShadingVs

f(V0)f(V0) f(V0) f(V0) f(V0) f(V0)

Shading

V0 V1

Gouroud ShadingVs

sf(V1)+(1-s)f(V0)f(V0) f(V1)




Geometric Modeling

Basic GLUT Shapes• Spheres

• Cones

• Cubes

glutWireSphere(radius, slices, stacks)

glutSolidSphere(radius, slices, stacks)

glutWireCone(radius, height, slices, stacks)

glutSolidCone(radius, height, slices, stacks)

glutWireCube(size)

glutSolidCube(size)

Basic GLUT Shapes• Torus

• Teapot

• Polyhedron

glutWireTorus(inner_radius, outer_radius, slices, stacks)

glutSolidTorus(inner_radius, outer_radius, slices, stacks)

glutWireTeapot(size)

glutSolidTeapot(size)

glutWireDodecahedron(void)

glutWireOctahedron(void)

glutWireTetrahedron(void)

glutWireIcosahedron(void)

Basic GLUT Shapes• GLUT shapes and normals are pre-defined.

• They are typically centered at the origin.

• To move them to suitable locations, you should apply transformations to them.

• Most of them do not have texture coordinates. (But the teapot has texture coordinates. Someone must really like the teapot…)

About the Teapot• It was created by Martin Newell at the University of Utah in

1975.

• It was the unofficial graphics mascot. (What is the current one?)

• Pixar gave away free teapot toys every year during ACM SIGGRAPH conference. Try to get one if you have chance.

Basic GLU Shapes• GLU also has some primitive shapes.

• The draw style: GL_FILL, GL_LINE, GL_SILHOUETTE, GL_POINT.

• The orientation: GL_OUTSIDE, GL_INSIDE.• Normal: GL_NONE, GL_FLAT, GL_SMOOTH.• Texture: GL_FALSE, GL_TRUE.

GLUquadric* quad= gluNewQuadric();gluQuadricDrawStyle(quad, draw);gluQuadricOrientation(quad, orientation);gluQuadricNormals(quad, normal);gluQuadricTexture(quad, texture);gluCylinder(quad, base, top, height, slices, stacks);gluDeleteQuadric(quad);

Basic GLU Shapes

• More complicated to use (than GLUT shapes).

• But they also have more options.

• Still not flexible enough to change many things.

Parametric Surfaces

• A sphere can be defined the polar coordinates/spherical coordinates (θ, ϕ).

xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

rsinθ sinφrcosφrcosθsinφ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

0 ≤θ ≤2π0 ≤φ ≤π

Creating a Sphere• The basic idea is to sample over (θ, ϕ).

2D Parameter Space 3D Space

θ

ϕ

(0, 0)

(2π, π)

(i, j) (i+1, j)

(i, j+1) (i+1, j+1)

Creating a Sphere• The code

for(i=0; i<n; i++)for(j=0; j<n; j++){

theta=i*2*PI/n;phi =j*PI/n;next_theta=(i+1)*2*PI/n;next_phi =(j+1)*PI/n;glBegin(GL_QUAD);glVertex3f(…i , j… );glVertex3f(…i+1, j… );glVertex3f(…i+1, j+1…);glVertex3f(…I , j+1…);glEnd();

}

Parametric Surfaces

• An ellipsoid is a scaled sphere.

xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

rx sinθsinφrycosφ

rz cosθsinφ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

0 ≤θ ≤2π0 ≤φ ≤π

Creating an Ellipsoid

• The basic idea is the same as creating a sphere, except that it uses different radii for different axes.

• It is wrong to create a sphere and then simply scale it. Why?

• Normal calculation is not straightforward. It needs more math, or some approximation.

Creating a Cylinder• The basic idea is to sample over (θ, h).

2D Parameter Space 3D Space

θ

h

(0, 0)

(2π, H)

(i, j) (i+1, j)

(i, j+1) (i+1, j+1)

Creating a Cylinder• The code

for(i=0; i<n; i++)for(j=0; j<n; j++){

theta=i*2*PI/n;phi =j*H/n;next_theta=(i+1)*2*PI/n;next_phi =(j+1)*H/n;glBegin(GL_QUAD);glVertex3f(…i , j… );glVertex3f(…i+1, j… );glVertex3f(…i+1, j+1…);glVertex3f(…I , j+1…);glEnd();

}

xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

rsinθhrcosθ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

0 ≤θ ≤2π0 ≤h≤H

linear interpolation

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