Fog

Fog is a useful effect for realistic large-scale scenes.

It provides an additional depth cue for distant objects - aerial perspective.

It can also be used to hide the edges of a 3D world.

glFog

OpenGL fog mixes the color of a fragment with the color of the fog. The degree of mixing is a function of the distance from the camera to the fragment.

finalColor = (1-f) * polyColor + f * fogColor

Example:

    glEnable(GL_FOG)
    glFogi(GL_FOG_MODE, GL_LINEAR)
    glFogfv(GL_FOG_COLOR, [1, 1, 1, 1])
    glFogf(GL_FOG_START, 5.0)
    glFogf(GL_FOG_END, 40.0)

Fog Mode

There are three different fog modes:

glFogi(GL_FOG_MODE, GL_LINEAR)

Fog ranges from none at near distance (GL_FOG_NEAR) to full at far distance (GL_FOG_FAR)

glFogi(GL_FOG_MODE, GL_EXP)

glFogi(GL_FOG_MODE, GL_EXP2)

Fog density is an exponential function of the distance. The density is varied by the GL_FOG_DENSITY parameter.

Interpolation

Linear: P = C₁t + C₀
C₁ = P₁ - P₀       C₀ = P₀

Cubic: P = C₃t³ + C₂t² + C₁t + C₀

Bezier Spline

Defined by 4 points.
Connects the first and last points; middle points control direction.

P(t) = (1 - t)³p0 + 3t(1 - t)²p1 + 3t²(1 - t)p2 + t³p3

Hermite Spline

Defined by 2 points, and tangents at those points

P(t) = (2t³ - 3t² + 1)p0 + (t³ - 2t² + t)m0 + (-2t³ + 3t²)p1 + (t³ - t²)m1

Catmull-Rom Spline

Defined by 4 points.
Curve passes through middle 2 points.

P = C₃t³ + C₂t² + C₁t + C₀

C₃ = -0.5 * P₀ + 1.5 * P₁ - 1.5 * P₂ + 0.5 * P₃
C₂ = P₀ - 2.5 * P₁ + 2.0 * P₂ - 0.5 * P₃
C₁ = -0.5 * P₀ + 0.5 * P₂
C₀ = P₁

Longer Paths

Linear interpolation & cubic curves generally connect only two points.

Often, we have many points that we want to connect with a single path

Piecewise Interpolation

A longer curve can be broken up into multiple pieces

Each piece is a line or curve connecting 2 points

For a Catmull-Rom curve, use the 4 surrounding points as control points

Piecewise Interpolation

The curve parameter (t) is a measure of how far along we are in the particular segment currently being interpolated

i.e. t=0 at the beginning point (P₁)
     t=1 at the end point (P₂)

Keyframe Animation

Keyframe animation uses a small sequence of key frames to define motion - all in-between frames are filled in based on the keys.

In computer animation, in-betweening is done by interpolation (linear or cubic)

Positions, rotation angles, scales, and other parameters can be interpolated.

This document is by Dave Pape, and is released under a Creative Commons License.