Nima Kalantari

CSCE 441 - Computer Graphics

​

Transformations 3D & Coordinates

Some slides from Ren Ng and Shinjiro Sueda; Some inspired by Steve Seitz

3D Transformations

• Use homogeneous coordinates again:
• 3D point = (x, y, z, 1)^T
• 3D vector = (x, y, z, 0)^T
• Use 4×4 matrices for affine transformations

3D Transformations

• Scale
• Translation

3D Transformations

• Rotation around x-, y-, or z-axis

Rotation �around �x-axis

3D Rotations

• Compose any 3D rotation from R_x, R_y, R_z?
• So-called Euler angles
• Often used in flight simulators: roll (x), pitch (y), yaw (z)
• Order matters!

z

y

x

Rodrigues’ Rotation Formula

• Rotation by angle α around axis n

Transformations

• Transform point and vectors
• E.g., move an object in the scene

​

• Transform coordinates
• E.g., transform coordinate of the driver from car coordinate to the world coordinate

​

How do we transform a mesh?

• Transform each one of the vertex positions by a transformation matrix

p₁

M

p’₁

p’₁ = Mp₁

How do we transform a mesh?

• Transform each one of the vertex positions by a transformation matrix

p₂

M

p’₂

p’₂ = Mp₂

How do we transform coordinates?

Coordinate Frames

• Any set of 3 vectors (in 3D) so that

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Coordinate Change: 2D Example

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​

​

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Coordinate Change: 2D Example

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​

​

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Coordinate Change: 2D Example

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​

​

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Coordinate Change: 2D Example

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​

​

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Coordinate Change: Extension to 3D

Chain coordinate transforms

1

2

3

4

Hierarchical transformation

Make a Robot Arm

Make a Robot Arm

• We start from a box
• Shrink vertically by 50%

Target

S

Make a Robot Arm

Target

S

Make a Robot Arm

Target

S

S

Make a Robot Arm

Target

S

S

T

Make a Robot Arm

Target

S

S

T

Make a Robot Arm

Target

S

S

T

S

Make a Robot Arm

Target

S

S

T

S

T

Make a Robot Arm

Target

S

S

T

S

T

R

Make a Robot Arm

Target

S

S

T

S

T

R

Make a Robot Arm

• Find a point you want to fix
• Move it to origin
• Then apply the transform
• Rotation in this case

​

Make a Robot Arm

Target

S

S

T

S

Make a Robot Arm

Target

S

S

T

S

T

Make a Robot Arm

Target

S

S

T

S

T

Make a Robot Arm

Target

S

S

T

S

T

R

Make a Robot Arm

Target

S

S

T

S

T

R

T

Make a Robot Arm

Target

S

S

T

S

T

R

T

Problems

• We need to come up with the matrices manually!!

TRTRTS

Problems

• Redo all the transformations!!

Hierarchical Modeling

Global Coordinate

Hierarchical Modeling

• Perform transformation in local (joint) coordinates!

Hierarchical Modeling

S^p

S^p

S^p

Hierarchical Modeling

T^p

T^p

T^p

S^p

S^p

S^p

Hierarchical Modeling

• How do we get the local (joint) coordinates?

S^p

S^p

S^p

T^p

T^p

T^p

Hierarchical Modeling

S^p

S^p

S^p

T^p

T^p

T^p

Hierarchical Modeling

S^p

S^p

S^p

T^p

T^p

T^p

R^c

Hierarchical Modeling

S^p

S^p

S^p

T^p

T^p

T^p

R^c

T^c

Hierarchical Modeling

• Opposite does not work!

S^p

S^p

S^p

T^p

T^p

T^p

R^c

T^c

Hierarchical Modeling

S^p

S^p

S^p

T^p

T^p

T^p

T^c

Hierarchical Modeling

S^p

S^p

S^p

T^p

T^p

T^p

T^c

R^c

Hierarchical Modeling

• Incorrect!

S^p

S^p

S^p

T^p

T^p

T^p

T^c

R^c

Hierarchical Modeling

S^p

S^p

S^p

T^p

T^p

T^p

R^c

T^c

Hierarchical Modeling

• Coordinate transformations are relative

S^p

S^p

S^p

T^p

T^p

T^p

R^c

T^c

Hierarchical Modeling

S^p

S^p

S^p

T^p

T^p

T^p

R^c

T^c

R^c

Hierarchical Modeling

S^p

S^p

S^p

T^p

T^p

T^p

R^c

T^c

R^c

T^c

Hierarchical Modeling

S^p

S^p

S^p

T^p

T^p

T^p

R^c

T^c

R^c

T^c

Hierarchical Modeling

S^p

S^p

S^p

T^p

T^p

T^p

R^c

T^c

R^c

T^c

R^c

T^c

Hierarchical Modeling

S^p

S^p

S^p

T^p

T^p

T^p

R^c

T^c

R^c

T^c

R^c

T^c

Part Transforms

Independent

Coord. Transforms

Children inherits parent’s transform

Effect of each term

S^p

S^p

S^p

T^p

T^p

T^p

R^c

T^c

R^c

T^c

R^c

T^c

Effect of each term

S^p

S^p

S^p

T^p

T^p

T^p

R^c

T^c

R^c

T^c

R^c

T^c

Effect of each term

S^p

S^p

S^p

T^p

T^p

T^p

R^c

T^c

R^c

T^c

R^c

T^c

Effect of each term

S^p

S^p

S^p

T^p

T^p

T^p

R^c

T^c

R^c

T^c

R^c

T^c

Effect of each term

S^p

S^p

S^p

T^p

T^p

T^p

R^c

T^c

R^c

T^c

R^c

T^c

Effect of each term

• R^c: rotation around joint

S^p

S^p

S^p

T^p

T^p

T^p

R^c

T^c

R^c

T^c

R^c

T^c

Effect of each term

• R^c: rotation around joint

S^p

S^p

S^p

T^p

T^p

T^p

R^c

T^c

R^c

T^c

R^c

T^c

Effect of each term

• R^c: rotation around joint

S^p

S^p

S^p

T^p

T^p

T^p

R^c

T^c

R^c

T^c

R^c

T^c

Effect of each term

• R^c: rotation around joint

S^p

S^p

S^p

T^p

T^p

T^p

R^c

T^c

R^c

T^c

R^c

T^c

Effect of each term

• R^c: rotation around joint

S^p

S^p

S^p

T^p

T^p

T^p

R^c

T^c

R^c

T^c

R^c

T^c

Effect of each term

• R^c: rotation around joint

S^p

S^p

S^p

T^p

T^p

T^p

R^c

T^c

R^c

T^c

R^c

T^c

Effect of each term

• R^c: rotation around joint

S^p

S^p

S^p

T^p

T^p

T^p

R^c

T^c

R^c

T^c

R^c

T^c

Effect of each term

• R^c: rotation around joint
• T^c: connection to parent

S^p

S^p

S^p

T^p

T^p

T^p

R^c

T^c

R^c

T^c

R^c

T^c

Effect of each term

• R^c: rotation around joint
• T^c: connection to parent

S^p

S^p

S^p

T^p

T^p

T^p

R^c

T^c

R^c

T^c

R^c

T^c

Effect of each term

• R^c: rotation around joint
• T^c: connection to parent

S^p

S^p

S^p

T^p

T^p

T^p

R^c

T^c

R^c

T^c

R^c

T^c

Effect of each term

• R^c: rotation around joint
• T^c: connection to parent

S^p

S^p

S^p

T^p

T^p

T^p

R^c

T^c

R^c

T^c

R^c

T^c

Effect of each term

• R^c: rotation around joint
• T^c: connection to parent

S^p

S^p

S^p

T^p

T^p

T^p

R^c

T^c

R^c

T^c

R^c

T^c

Effect of each term

• R^c: rotation around joint
• T^c: connection to parent

S^p

S^p

S^p

T^p

T^p

T^p

R^c

T^c

R^c

T^c

R^c

T^c

Effect of each term

• R^c: rotation around joint
• T^c: connection to parent

S^p

S^p

S^p

T^p

T^p

T^p

R^c

T^c

R^c

T^c

R^c

T^c

Effect of each term

• R^c: rotation around joint
• T^c: connection to parent
• S^p: shape of part

S^p

S^p

S^p

T^p

T^p

T^p

R^c

T^c

R^c

T^c

R^c

T^c

Effect of each term

• R^c: rotation around joint
• T^c: connection to parent
• S^p: shape of part

S^p

S^p

S^p

T^p

T^p

T^p

R^c

T^c

R^c

T^c

R^c

T^c

Effect of each term

• R^c: rotation around joint
• T^c: connection to parent
• S^p: shape of part

S^p

S^p

S^p

T^p

T^p

T^p

R^c

T^c

R^c

T^c

R^c

T^c

Effect of each term

• R^c: rotation around joint
• T^c: connection to parent
• S^p: shape of part
• T^p: location of joint

S^p

S^p

S^p

T^p

T^p

T^p

R^c

T^c

R^c

T^c

R^c

T^c

Effect of each term

• R^c: rotation around joint
• T^c: connection to parent
• S^p: shape of part
• T^p: location of joint

S^p

S^p

S^p

T^p

T^p

T^p

R^c

T^c

R^c

T^c

R^c

T^c

Effect of each term

• R^c: rotation around joint
• T^c: connection to parent
• S^p: shape of part
• T^p: location of joint

S^p

S^p

S^p

T^p

T^p

T^p

R^c

T^c

R^c

T^c

R^c

T^c

Effect of each term

• R^c: rotation around joint
• T^c: connection to parent
• S^p: shape of part
• T^p: location of joint

​

S^p

S^p

S^p

T^p

T^p

T^p

R^c

T^c

R^c

T^c

R^c

T^c

Effect of each term

• R^c: rotation around joint
• T^c: connection to parent
• S^p: shape of part
• T^p: location of joint

​

S^p

S^p

S^p

T^p

T^p

T^p

R^c

T^c

R^c

T^c

R^c

T^c

Effect of each term

• R^c: rotation around joint
• T^c: connection to parent
• S^p: shape of part
• T^p: location of joint

​

S^p

S^p

S^p

T^p

T^p

T^p

R^c

T^c

R^c

T^c

R^c

T^c

Effect of each term

• R^c: rotation around joint
• T^c: connection to parent
• S^p: shape of part
• T^p: location of joint

​

S^p

S^p

S^p

T^p

T^p

T^p

R^c

T^c

R^c

T^c

R^c

T^c

Effect of each term

• R^c: rotation around joint
• T^c: connection to parent
• S^p: shape of part
• T^p: location of joint

​

S^p

S^p

S^p

T^p

T^p

T^p

R^c

T^c

R^c

T^c

R^c

T^c

Putting it all together

• First, part

S^p

Putting it all together

• First, part

​

S^p

T^p

Putting it all together

• First, part

S^p

T^p

Putting it all together

• First, part
• Second, coordinate

S^p

T^p

R^c

Putting it all together

• First, part
• Second, coordinate

S^p

T^p

R^c

T^c

Putting it all together

• First, part
• Second, coordinate

S^p

T^p

R^c

T^c

Putting it all together

• First, part
• Second, coordinate

S^p

T^p

R^c

T^c

S^p

T^p

Putting it all together

• First, part
• Second, coordinate

S^p

T^p

R^c

T^c

S^p

T^p

Putting it all together

• First, part
• Second, coordinate

S^p

T^p

R^c

T^c

S^p

T^p

R^c

Putting it all together

• First, part
• Second, coordinate

S^p

T^p

R^c

T^c

S^p

T^p

R^c

T^c

Putting it all together

• First, part
• Second, coordinate

S^p

T^p

R^c

T^c

S^p

T^p

R^c

T^c

R^c

Putting it all together

• First, part
• Second, coordinate

S^p

T^p

R^c

T^c

S^p

T^p

R^c

T^c

R^c

T^c

Putting it all together

• First, part
• Second, coordinate

S^p

T^p

R^c

T^c

S^p

T^p

R^c

T^c

R^c

T^c

Putting it all together

• First, part
• Second, coordinate

S^p

T^p

R^c

T^c

S^p

T^p

R^c

T^c

R^c

T^c

S^p

T^p

Putting it all together

• First, part
• Second, coordinate

S^p

T^p

R^c

T^c

S^p

T^p

R^c

T^c

R^c

T^c

S^p

T^p

R^c

T^c

Putting it all together

• First, part
• Second, coordinate

S^p

T^p

R^c

T^c

S^p

T^p

R^c

T^c

R^c

T^c

S^p

T^p

R^c

T^c

Putting it all together

• First, part
• Second, coordinate

S^p

T^p

R^c

T^c

S^p

T^p

R^c

T^c

R^c

T^c

S^p

T^p

R^c

T^c

R^c

T^c

Putting it all together

• First, part
• Second, coordinate

S^p

T^p

R^c

T^c

S^p

T^p

R^c

T^c

R^c

T^c

S^p

T^p

R^c

T^c

R^c

T^c

Putting it all together

• First, part
• Second, coordinate

S^p

T^p

R^c

T^c

S^p

T^p

R^c

T^c

R^c

T^c

S^p

T^p

R^c

T^c

R^c

T^c

R^c

T^c

Putting it all together

• First, part
• Second, coordinate

S^p

T^p

R^c

T^c

S^p

T^p

R^c

T^c

R^c

T^c

S^p

T^p

R^c

T^c

R^c

T^c

R^c

T^c

Putting it all together

• First, part
• Second, coordinate

S^p

T^p

R^c

T^c

S^p

T^p

R^c

T^c

R^c

T^c

S^p

T^p

R^c

T^c

R^c

T^c

R^c

T^c

Robot – Assignment 2

• torso
• head
• right upper arm
• right lower arm
• left upper arm
• left lower arm
• right upper leg
• right lower leg
• left upper leg
• left lower leg

Robot – Assignment 2

• Parts connected at joints
• Each joint has 3 degrees of freedom
• Rotation around 3 axis (R^c)

Robot – Assignment 2

• Parts connected at joints
• Each joint has 3 degrees of freedom
• Rotation around 3 axis (R^c)

Torso

Head

R Upper Arm

L Upper Arm

R Upper Leg

L Upper Leg

R Lower Arm

L Lower Arm

R Lower Leg

L Lower Leg

T^p, S^p, T^c, R^c

User Controlled

Implementation using a stack

Stack

Push

Stack

Push

Push

Stack

Push

Push

Push

Stack

Push

Push

Push

Pop

Stack

Push

Push

Push

Pop

Pop

Stack

Push

Push

Push

Pop

Pop

Push

Matrix Stack for Graph Traversal

• The 1^st element is the identity matrix

Matrix Stack for Graph Traversal

• The 1^st element is the identity matrix

I

Matrix Stack for Graph Traversal

• The 1^st element is the identity matrix
• Push:
• Copy the previous top matrix
• Push it to the stack

I

I

I

Push

Matrix Stack for Graph Traversal

• The 1^st element is the identity matrix
• Push:
• Copy the previous top matrix
• Push it to the stack
• Pop:
• Remove the top element

I

I

I

…

A

…

A

A

Push

Push

…

Matrix Stack for Graph Traversal

• Each element in stack is an array of 16 numbers
• Corresponding to the elements of a 4x4 matrix

I

I

I

…

A

…

A

A

Push

Push

…

Matrix Stack for Graph Traversal

• New matrices are right-multiplied to the top matrix
• translate(), scale(), rotate(), etc.

…

A

…

Matrix Stack for Graph Traversal

• New matrices are right-multiplied to the top matrix
• translate(), scale(), rotate(), etc.

…

A

…

A

A

Push

…

Matrix Stack for Graph Traversal

• New matrices are right-multiplied to the top matrix
• translate(), scale(), rotate(), etc.

…

A

…

A

A

Push

…

…

A

A T

Translate

Matrix Stack for Graph Traversal

• New matrices are right-multiplied to the top matrix
• translate(), scale(), rotate(), etc.

…

A

…

A

A

Push

…

…

A

A T

Translate

…

A

A T S

Scale

Matrix Stack for Graph Traversal

• New matrices are right-multiplied to the top matrix
• translate(), scale(), rotate(), etc.

…

A

…

A

A

Push

…

…

A

A T

Translate

…

A

A T S

Scale

…

A

A T S R

Rotate

Matrix Stack for Graph Traversal

• New matrices are right-multiplied to the top matrix
• translate(), scale(), rotate(), etc.

…

A

…

A

A

Push

…

…

A

A T

Translate

…

A

A T S

Scale

…

A

A T S R

Rotate

…

A

Pop

Robot

Robot

Robot

• Depth first traversal

I

Stack

T^c

R^c

T^p

S^p

T^c

R^c

T^p

S^p

T^c

R^c

T^p

S^p

T^c

R^c

T^p

S^p

T^c

R^c

T^p

S^p

Robot

• Depth first traversal

T^c

R^c

T^p

S^p

T^c

R^c

T^p

S^p

T^c

R^c

T^p

S^p

T^c

R^c

T^p

S^p

T^c

R^c

T^p

S^p

I

Stack

Robot

• Depth first traversal

T^c

R^c

T^p

S^p

T^c

R^c

T^p

S^p

T^c

R^c

T^p

S^p

T^c

R^c

T^p

S^p

T^c

R^c

T^p

S^p

I

Stack

Robot

• Depth first traversal

T^c

R^c

T^p

S^p

T^c

R^c

T^p

S^p

T^c

R^c

T^p

S^p

T^c

R^c

T^p

S^p

T^c

R^c

T^p

S^p

I

Stack

Robot

• Depth first traversal

T^c

R^c

T^p

S^p

T^c

R^c

T^p

S^p

T^c

R^c

T^p

S^p

T^c

R^c

T^p

S^p

T^c

R^c

T^p

S^p

I

Stack

Robot

• Depth first traversal

T^c

R^c

T^p

S^p

T^c

R^c

T^p

S^p

T^c

R^c

T^p

S^p

T^c

R^c

T^p

S^p

T^c

R^c

T^p

S^p

I

Stack

Robot

• Depth first traversal

T^c

R^c

T^p

S^p

T^c

R^c

T^p

S^p

T^c

R^c

T^p

S^p

T^c

R^c

T^p

S^p

T^c

R^c

T^p

S^p

I

Stack

Recap

• First, part
• Second, coordinate

S^p

T^p

R^c

T^c

S^p

T^p

R^c

T^c

R^c

T^c

S^p

T^p

R^c

T^c

R^c

T^c

R^c

T^c

Robot

I

Stack

C

P

C

P

C

P

C

P

C

P

Robot

C

P

C

P

C

P

C

P

C

P

I

Stack

Robot

I

Stack

Push()

Pop()

I

C

P

C

P

C

P

C

P

C

P

Robot

I

Stack

C

Push()

Coordinate transform(C)Pop()

C

P

C

P

C

P

C

P

C

P