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Geometry: �Coordinates & Cameras

Instructor: Christopher Rasmussen (cer@cis.udel.edu)

February 15, 2022 ❖ Lecture 3

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Outline

Coordinate systems
More 3-D transformations

Setting up the camera

Projections

Orthographic
Perspective

Marschner readings: 6.5-7.2

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Coordinate Systems

What does it mean to say a point x = (x, y, z) is in a certain coordinate system or frame?
One way to think of x is as a weighted sum (aka linear combination) of basis vectors representing the frame’s axes
So x = x(1, 0, 0) + y(0, 1, 0) + z(0, 0, 1) in the local frame

x

Z

X

Y

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Coordinate Systems

If the frame of interest is embedded in another frame F, then its origin and axes can be represented in F as a point o and unit vectors i, j, and k, respectively

x

k

i

j

F

o

Z

X

Y

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Coordinate Systems

If the frame of interest is embedded in another frame F, then its origin and axes can be represented in F as a point o and unit vectors i, j, and k, respectively

These vectors can have any orientation in F, but are all orthogonal to each other
Later we will also refer to these with u, v, n or

u, v, w (sorry!)

x

w

(or n)

u

v

F

o

Z

X

Y

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Coordinate Systems

Now x, which in the local coordinate frame is just (x, y, z), can be thought of in F as:

x = o + xi + yj + zk

x

k

i

j

F

o

Z

X

Y

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Coordinate Systems

Now x, which in the local coordinate frame is just (x, y, z), can be thought of in F as:

x = o + xi + yj + zk

x

k

i

j

F

o

Z

X

Y

In Marschner: p + uu + vv + ww

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3-D Transformations:� Arbitrary Change of Coordinates

A rigid transformation is used to represent a change in the coordinate system that “expresses” a point’s location

Y_A

X_A

Z_A

A

x

Z_B

X_B

Y_B

B

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3-D Transformations:� Arbitrary Change of Coordinates

A rigid transformation is used to represent a change in the coordinate system that “expresses” a point’s location

Y_A

Z_A

A

Z_B

X_B

Y_B

B

X_A

^Ax

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3-D Transformations:� Arbitrary Change of Coordinates

A rigid transformation is used to represent a change in the coordinate system that “expresses” a point’s location

Y_A

Z_A

A

Z_B

X_B

Y_B

B

X_A

^Bx

See Fig. 6.20 in Marschner for 2-D equivalent

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3-D Transformations:� Arbitrary Change of Coordinates

A rigid transformation is used to represent a change in the coordinate system that “expresses” a point’s location

A

B

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3-D Transformations:� Arbitrary Change of Coordinates

A rigid transformation is used to represent a change in the coordinate system that “expresses” a point’s location

A

B

M transforms x from A's to B's coordinate system

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3-D Transformations:� Translation-only Change of Coordinates

A rigid transformation is used to represent a change in the coordinate system that “expresses” a point’s location

Location of A’s origin in

B’s coordinate system

A

B

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3-D Transformations:� Arbitrary Change of Coordinates

A rigid transformation is used to represent a change in the coordinate system that “expresses” a point’s location

Rotation that makes

A’s axes parallel to B’s,

creating translation-

only case

A

B

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3-D Transformations:� Arbitrary Change of Coordinates

A rigid transformation is used to represent a change in the coordinate system that “expresses” a point’s location

A

B

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3-D Rigid Transformations

Combination of rotation followed by translation, without scaling, etc.
“Moves” an object from one 3-D pose to another

T

R

M

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Rigid Transformations: Homogeneous Coordinates

Points in one coordinate system are transformed to the other as follows:

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Rigid Transformations: Homogeneous Coordinates

Points in one coordinate system are transformed to the other as follows:

same as frame-to-canonical

matrix in Marschner 6.5

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Rigid Transformations: Homogeneous Coordinates

Points in one coordinate system are transformed to the other as follows:

Rows of rotation matrix are B ’s axes “in” A ’s coordinate system

same as frame-to-canonical

matrix in Marschner 6.5

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Rigid Transformations: Homogeneous Coordinates

Points in one coordinate system are transformed to the other as follows:

Rows of rotation matrix are B ’s axes “in” A ’s coordinate system
^C_WM transforms points expressed in world coordinates ^Wx into points expressed in camera coordinates ^Cx. Equivalently, it transforms a camera object to the world origin

Info needed for ^C_WM: (1) Camera axes in world coordinates, (2) world origin in camera coordinates

same as frame-to-canonical

matrix in Marschner 6.5

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Rigid Transformations: Homogeneous Coordinates

Points in one coordinate system are transformed to the other as follows:

Rows of rotation matrix are B ’s axes “in” A ’s coordinate system
^C_WM transforms points expressed in world coordinates ^Wx into points expressed in camera coordinates ^Cx. Equivalently, it transforms a camera object to the world origin

Info needed for ^C_WM: (1) Camera axes in world coordinates, (2) world origin in camera coordinates

same as frame-to-canonical

matrix in Marschner 6.5

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Rigid Transformations: Homogeneous Coordinates

Points in one coordinate system are transformed to the other as follows:

Rows of rotation matrix are B ’s axes “in” A ’s coordinate system
^C_WM transforms points expressed in world coordinates ^Wx into points expressed in camera coordinates ^Cx. Equivalently, it transforms a camera object to the world origin

Info needed for ^C_WM: (1) Camera axes in world coordinates, (2) world origin in camera coordinates

same as frame-to-canonical

matrix in Marschner 6.5

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Controlling the camera position

Blender: camera pose + object, right-handed coordinates. Origin of camera coordinate system C at tip of pyramid, note "up" arrow. World coordinates W coincide with cube object centroid (aka "model" coordinates M)

Similar view in Unity, left-handed coordinates. Here camera axes are explicit and world coordinates are not same as model coordinates.

RGB axis colors correspond to XYZ ordering

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Transformations in Blender

Change view by clicking/dragging axes in scene
Zoom in/out, move view, toggle user/camera

Select object in scene or Scene Collection pane

Click and drag with Move, Rotate, and Scale options in scene
Click Transform drop-down, directly edit Location/Rotation/Scale

Viewport Navigation in Blender tutorial

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Transformations in OpenGL

A variety of transformations were built into OpenGL before 3.0 --- you defined a vertex v = (x, y, z) and called a function like glTranslate() to transform it

All this did was create a transformation matrix to be multiplied

Now the programmer is "responsible" for building transformation matrices
Luckily, other libraries exist to make this easy; we will use GLM (OpenGL Mathematics)

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Controlling the camera position in OpenGL

Standard OpenGL frame: At (0, 0, 0)^T in world coordinates looking in -Z direction (0, 0, -1) with up vector (0, 1, 0)^T

Up vector controls camera roll (rotation around Z axis)

Changing position: "glm::l ook A t" function

eye = (eyeX, eyeY, eyeZ)^T: Desired camera position
center = (centerX, centerY, centerZ)^T: Point at which camera is aimed (defining “gaze direction”, aka camera pitch and yaw)
up = (upX, upY, upZ)^T: Camera’s “up” vector

from Woo et al.

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glm::lookAt: What It Does

Creates rigid transformation ^C_WM making a change from world to camera coordinates
Commonly called the view matrix V
Effect is to move scene points ^Wx to ^Cx so that camera is at origin, “look at” point is on -Z axis, and camera +Y axis is aligned with up vector
Up to programmer to apply it in the right place

More details in Marschner, Chapter 7.1.3

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glm::lookAt: How It Works

Compute vectors u, v, n defining new camera axes in world coordinates (Marschner textbook uses w instead of n)

“Old” axes are u’ = (1, 0, 0)^T, v’ = (0, 1, 0)^T, n’ = (0, 0, 1)^T

Compute location ^Co_W of old camera position (i.e., world origin) in terms of new location’s coordinate system
Fill in rigid transform matrix

from Hill

center

C

W

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glm::lookAt: Camera axes in world coords.

Form basis vectors

New camera Z axis: ñ = eye - center
New camera X axis: ũ = up x n
New camera Y axis: v = n x u (not necessarily same as up)

Normalize to make unit vectors: n = ñ/‖ñ‖, etc.

from Hill

center

C

W

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glm::lookAt: Camera axes in world coords.

Form basis vectors

New camera Z axis: ñ = eye - center
New camera X axis: ũ = up x n
New camera Y axis: v = n x u (not necessarily same as up)

Normalize to make unit vectors: n = ñ/‖ñ‖, etc.

from Hill

center

C

W

ñ

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glm::lookAt: Camera axes in world coords.

Form basis vectors

New camera Z axis: ñ = eye - center
New camera X axis: ũ = up x n
New camera Y axis: v = n x u (not necessarily same as up)

Normalize to make unit vectors: n = ñ/‖ñ‖, etc.

from Hill

center

up

C

W

up and n define a plane

which ũ is normal to—they

don’t have to be orthogonal

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glm::lookAt: Camera axes in world coords.

Form basis vectors

New camera Z axis: ñ = eye - center
New camera X axis: ũ = up x n
New camera Y axis: v = n x u (not necessarily same as up)

Normalize to make unit vectors: n = ñ/‖ñ‖, etc.

from Hill

center

C

W

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glm::lookAt: Camera axes in world coords.

Form basis vectors

New camera Z axis: ñ = eye - center
New camera X axis: ũ = up x n
New camera Y axis: v = n x u (not necessarily same as up)

Normalize to make unit vectors: n = ñ/‖ñ‖, etc.

from Hill

center

C

W

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glm::lookAt: Camera axes in world coords.

Form basis vectors

New camera Z axis: ñ = eye - center
New camera X axis: ũ = up x n
New camera Y axis: v = n x u (not necessarily same as up)

Normalize to make unit vectors: n = ñ/‖ñ‖, etc.

from Hill

center

C

W

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glm::lookAt: Camera axes in world coords.

Now make 3 x 3 rotation matrix as described on rigid transform slide:

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glm::lookAt: Location

: World origin in camera coordinates

center

from Hill

origin

C

W

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glm::lookAt: Location

: World origin in camera coordinates

center

from Hill

origin

C

W

eye

eye is camera origin in world coordinates -- aren't they just the reverse of each other?

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glm::lookAt: Location

: World origin in camera coordinates

center

from Hill

origin

C

W

-eye

eye is camera origin in world coordinates -- aren't they just the reverse of each other?

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glm::lookAt: Location

: World origin in camera coordinates
-eye is in world coordinates, so project onto camera axes (and don’t normalize):

from Hill

origin

-eye

C

W

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glm::lookAt: Matrix

Letting and writing the vector components as u = (u_x, u_y, u_z)^T, etc., the final transformation matrix is given by:

V =

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Transformations vs. Projections in CG

Transformation: Mapping within n-D space that “moves points around”

E.g., 2-D and 3-D rotation, translation, scaling, shear, but also nonlinear distortions

Projection: Mapping from n-D space down to lower-dimensional subspace

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Transformations in CG

Mapping within n-D space that moves points around

Linear transformations (= matrix multiplication) preserve straight lines
Some nonlinear transformations in n-D can be expressed by linear ones in (n + 1)-D – the idea behind homogeneous coordinates

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Projections in CG

Mapping from n-D space down to lower-dimensional subspace

E.g., point in 3-D space to point on plane (a 2-D entity) in that space
We will be interested in such 3-D to 2-D projections where the plane is the image plane

Things to know:

Where is the plane? Derived from view matrix V
What region of the plane does the image occupy?
What kind of projection?

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The Viewing Volume

Definition: The region of 3-D space visible in the image
Depends on:

Camera position,
Camera orientation
Field of view
Image size
Projection type

Orthographic
Perspective

courtesy of N. Robins

Pyramid-shaped viewing volumes are characteristic of perspective projection -- more on Thursday