Viewing for 3D Graphics

Viewing for 3D Graphics
An Introduction

Prof. David Bernstein
James Madison University

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?