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Geometry: �Coordinates & Cameras

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

Course web page:

https://goo.gl/th9HB2

February 13, 2018 ❖ 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

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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

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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!)

(or n)

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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

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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

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

Z_B

X_B

Y_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

Z_B

X_B

Y_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

Z_B

X_B

Y_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

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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

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

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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

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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

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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

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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 takes the camera to the world origin, transforming points expressed in world coordinates into points expressed in camera coordinates

Info needed for M: (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

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”)
up = (upX, upY, upZ)^T: Camera’s “up” vector

from Woo et al.

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glm::lookAt: What It Does

Moves scene points so that camera is at origin, “look at” point is on -Z axis, and camera +Y axis is aligned with up vector
Create and execute rigid transformation ^C_WM making a change from world to camera coordinates

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 in terms of new location’s coordinate system
Fill in rigid transform matrix

from Hill

center

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glm::lookAt: Camera axes in world coords.

Form basis vectors

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

Normalize so that these are unit vectors

from Hill

center

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glm::lookAt: Camera axes in world coords.

Form basis vectors

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

Normalize so that these are unit vectors

from Hill

center

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glm::lookAt: Camera axes in world coords.

Form basis vectors

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

Normalize so that these are unit vectors

from Hill

center

up and n define a plane

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

Normalize so that these are unit vectors

from Hill

center

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glm::lookAt: Camera axes in world coords.

Form basis vectors

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

Normalize so that these are unit vectors

from Hill

center

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glm::lookAt: Camera axes in world coords.

Form basis vectors

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

Normalize so that these are unit vectors

from Hill

center

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glm::lookAt: Camera axes in world coords.

Now make 3 x 3 rotation matrix from formula on rigid transform slide:

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glm::lookAt: Location

: World origin in camera coordinates

from Hill

center

origin

-eye

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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

center

origin

-eye

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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:

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Transformations vs. Projections

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

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

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
Things to know:

Where are the points?
Where is the plane?
What kind of projection?

Parallel projection along direction d onto a plane

from Hill

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Parallel Projections

Oblique: d in general position

relative to image plane normal n

Orthographic: d parallel to n

from Hill

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The Viewing Volume

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

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

Orthographic
Perspective

courtesy of N. Robins

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Orthographic Projection

Projection direction d is aligned with Z axis
Viewing volume is “brick”-shaped region in space

Not the same as image size

No perspective effects—distant objects look same as near ones, so camera (x, y, z) ⇒ image (x, y)

from Hill

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Camera Orientation under Orthographic Projection (set by lookat)

Plan: Projection onto XZ (ground) plane
Elevations

Front: Projection onto XY plane
Side: Projection onto YZ plane

courtesy of http://glasnost.itcarlow.ie/~powerk/GeneralGraphicsNotes/projection/orthographicprojection.html

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Camera Orientation under Orthographic Projection (set by lookat)

Isometric: "¾ view" -- axes equally scaled
Dimetric: 2 axes the same, 1 different
Trimetric: All axes different

By SharkD - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=8497328

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Camera Orientation under Orthographic Projection (set by lookat)

Isometric: "¾ view" -- axes equally scaled
Dimetric: 2 axes the same, 1 different
Trimetric: All axes different

some images courtesy of http://glasnost.itcarlow.ie/~powerk/GeneralGraphicsNotes/projection/orthographicprojection.html

SimCity 2000 (not quite isometric)

Diablo