Viewing and Projections Dr. Amy Zhang. Reading 2 Hill, Chapter 5 and 7 Red Book, Chapter 3, “Viewing”

Viewing and Projections

Dr. Amy Zhang

Reading

2

Hill, Chapter 5 and 7 Red Book, Chapter 3, “Viewing”

3D Graphics Pipeline

3

The big picture…

Outline

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Camera Models Viewing Transformation Projection Matrix OpenGL Transformation Pipeline

Cameras

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Cameras have an optical system: Filters Lenses Aperture

The projection surface may be flat or curved, oriented at various angles with respect to the incoming light.

Examples: A camera or the eye.

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Camera Obscura

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The first cameras – a dark box with a small hole in it

The Pinhole Camera

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An abstract camera model Models the geometry of perspective projection Used in most of computer graphics

Pinhole Optics

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Perspective

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Perspective Derivation

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Consider the projection of a point onto the projection plane:

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By similar triangles we can compute how much the x and y-coordinates are scaled

Looking down y axis:

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We get:

This is clearly a non-linear transformation BUT: We can split it into a linear part followed

by a nonlinear part

Homogeneous Coordinates

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Remember homogeneous coordinates:

To get a homogeneous point we divide all the coordinates by w:

This is called the perspective divide

Perspective Projections

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We can now rewrite the perspective projection as a linear transformation:

After division by the 4th component we get:

The Reason for Lenses *

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The Lens Model

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Lens, aperture, and image plane

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Focal length f: the distance from lens to image plane

A point in focus: the image of a point is on the image plane

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An out-of-focus point The circle of confusion r

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The Gaussian / thin lens formula:

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A near point in out-of-focus

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The depth of focus dfocus and the depth of field (DOF) dfield

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Decreasing the aperture size reduces the size of the blur for points not in the focused plane, so that the blurring is imperceptible, and all points are within the dfield.

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Viewing and Projection

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In OpenGL we distinguish between: Viewing: placing the camera Projection: describing the viewing frustum of the

camera (and thereby the projection transformation)

Perspective divide: computing homogeneous points

Outline

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OpenGL Transformations

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The viewing transformation V transforms a point from world space to eye space:

Placing the Camera

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It is most natural to position the camera in world space as if it were a real camera

Identify the eye point where the camera is located

Identify the look-at point that we wish to appear in the center of our view

Identify an up-vector vector that we wish to be oriented upwards in our final image

Look-At Positioning

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We specify the view frame using the look-at vector a and the camera up vector up

The vector a points in the negative viewing direction

In 3D, we need a third vector that is perpendicular to both up and a to specify the view frame

Where does it point to?

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The result of the cross product is a vector, not a scalar, as for the dot product

Depending on the basis vectors i, j, and k, the new vector follows the right or left handed rule

In OpenGL, the cross product a x b yields a right hand side (RHS) vector perpendicular to a and b

Computing Cross Products

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We can compute the cross product using yet another matrix-vector multiplication:

The matrix is sometimes called the skew-symmetric matrix of the vector (in this case a)

Cross products produce vectors for both vector and point inputs

Constructing a Frame

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The cross product between the up and the look-at vector will get a vector that points to the right.

Finally, using the vector a and the vector r we can synthesize a new vector u in the up direction:

World and Camera Frames

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The relation between the world and the camera is expressed as:

We move the eye (camera) by updating E

Rotation

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Rotation first:

Translation

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Translation to the eye point:

Composing the Result

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The final camera transformation is:

Why?

The Viewing Transformation

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Expressing P in eye coordinates:

The Viewing Transformation

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As a single 4x4 matrix:

Where these are normalized vectors:

gluLookAt()

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OpenGL provides a very helpful utility function that implements the look‐at viewing specification:

These parameters are expressed in world coordinates

Outline

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The projection transformation P transforms a point from eye space to clip space:

Projection Transformations

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Projections fall into two categories: Parallel projections: The camera is placed at an

infinite distance from the viewplane; lines of projection are parallel to each other

Perspective projections: Lines of projection converge at a point

Parallel Projections

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The simplest form of parallel projection is simply along lines parallel to the z-axis onto the xy-plane

This form of projection is called orthographic

For other parallel projections see, e.g.:http://www.mtsu.edu/~csjudy/planeview3D/tutorialparallel.html

Orthographic Frustum

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The user specifies the orthographic viewing frustum by specifying minimum and maximum x/y coordinates

It is necessary to indicate a range of distances along the z-axis by specifying near and far planes

Orthographic Projections to NDC

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Normalized Device Coordinate (NDC) makes up a coordinate system that describes positions on a virtual plotting device

Here is the orthographic world-to-clip transformation: Move center to origin

T(-(left+right)/2, -(bottom+top)/2,(near+far)/2)) Scale to have sides of length 2

S(2/(right-left),2/(top-bottom),2/(far-near))

1000

200

02

0

002

nearfar

nearfar

nearfar

bottomtop

bottomtop

bottomtop

leftright

leftright

leftright

P = ST =

Orthographic Projection in OpenGL

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This matrix is constructed with the following OpenGL call:

And the 2D version (another GL utility function):

Just a call to glOrtho() with near = -1 and far = +1

Properties of Parallel Projections

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Not realistic looking Good for exact measurements A kind of affine transformation

Parallel lines remain parallel Ratios are preserved Angles (in general) not preserved

Most often used in CAD, architectural drawings, etc., where taking exact measurement is important

Isometric Games

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A special kind of parallel projection called isometric projection is often used in games

It’s essentially a shear and an orthographic projection

Easier to compute than a full perspective transformation

SimCity

The Sims

Diablo

Perspective Projections

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Artists (Donatello, Brunelleschi, and Da Vinci) during the renaissance discovered the importance of perspective for making images appear realistic

Parallel lines intersect at a point

Perspective Viewing Frustum

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Just as in the orthographic case, we specify a perspective viewing frustum

Values for left, right, top, and bottom are specified at the near depth.

Perspective Viewing Frustum

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OpenGL provides a function to set up this perspective transformation:

There is also a simpler OpenGL utility function:

fov = vertical field of view in degrees aspect = image width / height at near depth

Can only specify symmetric viewing frustums where the viewing window is centered around the –z axis.

OpenGL Clip Space

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In OpenGL clip space, the viewing frustum is mapped to a cube that extends from -1 to 1 in x, y, and z.

OpenGL also flips the z axis to create a left handed coordinate system during projection

OpenGL Perspective Matrix

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Mapping the perspective viewing frustum in OpenGL to clip space involves some affine transformations

OpenGL uses a clever composition of these transformations with the perspective projection matrix:

What happens to the z value?

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We get the following expression for z from our perspective matrix:

This mapping is non‐linear:

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Uniform differences in 3D (z) depth values do not correspond to uniform differences in z’ values:

The number of discernable depths is greater near the near plane than the far plane Tip: Choose the near plane as far away as

possible

Properties of Perspective Projections

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The perspective projection is an example of a projective transformation

Here are some properties of projective transformations: Lines map to lines Parallel lines do not necessarily remain parallel Ratios are not preserved

One of the advantages of perspective projection is that size varies inversely with distance – looks realistic

A disadvantage is that we can't judge distances as exactly as we can with parallel projections

Outline

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The rest of the OpenGL transformation pipeline:

3D:

2D:

Clipping & Perspective Division

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The scene's objects are clipped against the clip space bounding box

This step eliminates any objects (and pieces of objects) that are not visible in the image

Hill describes an efficient clipping algorithm for homogeneous clip space

Perspective division divides all homogeneous coordinates through w

Clip space becomes Normalized Device Coordinate (NDC) space after the perspective division

Viewport Transformation

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One last transformation from NDC space to the viewport on the screen

The viewport transformation also maps NDC depth values from the range [-1,1] to [0, 1]

The resulting screen space is still 3D with depth values between 0 and 1

Viewport Transformation

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OpenGL provides a function to set up the viewport transformation:

Screen Coordinate Systems

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The rasterization step scan converts the object into pixels


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A z-buffer depth test resolves visibility of the objects on a per-pixel basis and writes the pixels to the frame buffer

The end

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Questions and answers

Documents

Viewing and Projections Dr. Amy Zhang. Reading 2 Hill, Chapter 5 and 7 Red Book, Chapter 3, “Viewing”