Viewing for 3D Graphics

Viewing for 3D Graphics
An Introduction

Computer Science Department

bernstdh@jmu.edu

Introduction

An Important Issue:
- Determining what part of a scene is visible
Gainining Some Intuition:
- A camera only takes a picture of part of the scene
A Common Approach:
- Use the concept of a pinhole camera (rather than, say, a single lens reflex camera) but put the "film" (called the view plane or screen) in front of the pinhole

The Axis-Aligned Case

Visualization:
Notation:
- $\bs{u}$ , $\bs{v}$ , and $\bs{w}$ denote the coordinate axes (i.e., the basis) for the view
- $\bs{e}$ denotes the eye/pinhole position (in scene/world coordinates) which is the origin in view coordinates
- $\bs{c}$ denotes the lower-left corner of the view plane
Notes:
- Nothing between the eye and the view plane is visible
- The "chopped pyramid" shape that contains everything that can be seen is a frustum (the walls of which are the clipping planes)

The General Case

Desirable Functionality:
- The ability to position the eye/camera anywhere and have it point in any direction
Visualization (when Centered):
Notation:
- $\bs{g}$ denotes the gaze direction (which is normal to the view plane)
- $\bs{h}$ denotes the head-up direction

The General Case (cont.)

What's Needed:
- A basis for the view plane (with $\bs{e}$ at the origin)
Determining \bs{w}:
- The opposite direction of $\bs{g}$ , and with unit length. So, $\bs{w} = -\frac{\bs{g}}{||\bs{g}||}$ .
Determining \bs{u}:
- Orthogonal to both $\bs{h}$ and $\bs{w}$ , and with unit length. So, $\bs{u} = -\frac{\bs{h} \times \bs{w}} {||\bs{h} \times \bs{w}||}$ .
Determining \bs{v}:
- Orthogonal to $\bs{w}$ and $\bs{u}$ , and with unit length. So, $\bs{v} = \bs{w} \times \bs{u}$ (which has unit length).

The General Case (cont.)

Another Consideration:
- The shape of the pixels
For Square Pixels:
- Must ensure that $\frac{2u}{2v} = \frac{u}{v}$ equals the aspect ratio (i.e., the width divided by the height)

Specifying Parameters

An Observation:
- There are many different ways to specify the parameters of this model
A Common Approach:
- Use the focal length (i.e., the distance between the eye/camera and the view plane), the vertical field of view (which is an angular measure), and the aspect ratio

Specifying Parameters (cont.)

A Visualization:
Interpretation:
- We are looking at the view plane "edge on" (so $\bs{u}$ is pointing out of the illustration)
Notation:
- $\phi$ denotes the (vertical) field of view
- $s$ denotes the focal length
Use:
- The "half height" can be calculated from $\phi$ and $s$
- The "half width" can be calculated from the "half height" and the aspect ratio

What's Missing?

There's Always More to Learn