interpolation in 3d computer graphics

Question

I was wondering if someone could help with explaining in simple terms what interpolation is and how its used in 3d computer graphics

Answer 1

Simply put: given two points A and B, find a point between them.

For example, if I want to move something along a line from a position x=1 to x=4 in one step:

1-----------------------4

The first step is at location 1, the second step is at location 4, so the object moves instantly from one location to the other. However, if I want the object to take a certain amount of time or number of frames to make the transition, I'll need to refine that by finding intermediate points that are evenly spaced.

If I want the object to take two steps (or frames) to move from 1 to 4,

1-----------X-----------4

I need to calculate what the new point (X) is so I can draw the object there at the appropriate time. In this case, the point X will be

                                  (max-min)
location = min + (current_step) * --------
                                    steps

location is what we're trying to find. min=1, max=4, and in this example steps=2 since we want to divide the span into two steps:

step:   location:
0       1
1       2.5
2       4

1------------(2.5)-----------4

If we want to take 4 steps:

step:   location:
0       1
1       1.75
2       2.5
3       3.25
4       4

1---(1.75)---(2.5)---(3.25)---4

And so forth. For four steps, the object moves 25% of the total distance per frame. For 10 steps, 10%, etc ad nauseum.

For multiple dimensions (when an object has a 2- or 3-dimensional trajectory), just apply this to each X,Y,Z axis independently.

This is linear interpolation. There are other kinds. As always, Google can help you out.

Other applications include texture mapping, anti-aliasing, image smoothing and scaling, etc., and of course many other uses outside of games and graphics.

Note: a lot of frameworks already provide this. In XNA, for instance, it's Matrix.Lerp.

Answer 2

Interpolation is the smooth adjustment from one thing to another. It is used in animation.

For example, if an object is at location 1, and we want to move it to location 2 over the course of six seconds, we need to slowly interpolate its location between the two endpoints. Interpolation also refers to any search for a location on that path.

Answer 3

Interpolation is the 'guessing' of points based on other points.

for example when you have the points (0,0) and (2,2) you might 'guess' that the point (1,1) also belongs to the set.

The simples application is to deduce a line from two points.

The same thing works in 3 or actually n-dimension.

In 3D graphics it will be used

• for animations, to calculate the position of things based on start and end coordinations
• calculating lines
• gradients
• scaling of graphics and probably many more

Answer 4

General Definition

Interpolation (in mathematics) can be regarded as a transition from one value to another. Interpolation usually uses a value in the 0 to 1 range like a percentage. 0 is the starting value and 1 is the end value. The main purpose of interpolation is to find values in between given values.

Types of Interpolation

There are many types of interpolation used in various programs, the most common being linear interpolation. This type of interpolation is the most simple and straight-forward; It is used to find values in a line segment between two points or numbers. There are also: cubic interpolation, quadratic interpolation, bilinear, trilinear, etc. For more information go here: https://en.wikipedia.org/wiki/Interpolation.

Application in 3D Graphics

Interpolation, especially linear, bilinear and trilinear, is important for computing fragments in geometry (the textures and visuals of the geometry), blending volumetric textures, mip-mapping (a depth of field effect on texture), and lighting (like unreal engine's volumetric lightmaps). The results of the interpolation may vary, but it could potentially yield very realistic results. It is a rather large computation, especially when the interpolation is in 3-dimensions or above (hyperspace).

Example of Interpolation

In 1 Dimension:

n1 = 1
n2 = 2

i = 0.5

n3 = (n1 - n1 * i) + n2 * i

///////////////////////////////////////

                  n3
├────────┼────────┼────────┼────────┤
1       1.25     1.5      1.75      2

///////////////////////////////////////

In 2 Dimensions:

v1 = {1, 1}
v2 = {1.5, 2}

i = 0.5
d = √((v1.x - v2.x)^2 + (v1.y - v2.y)^2)

v3 = {v1.x + -d * i * ((v1.x - v2.x) / d),v1.y + -d * i * ((v1.y - v2.y) / d)} 

///////////////////////////////

  2 ┼─┼─┼─┼─┼─┼─┼─┼─┼─┼─┼─┼─┼
    ┼─┼─┼─┼─┼─┼─┼─┼─┼─┼─┼─┼─┼
    ┼─┼─┼─┼─┼─┼─┼─┼─┼─┼─┼─┼─┼
    ┼─┼─┼─┼─┼─┼─┼─┼─┼─┼─┼─┼─┼
    ┼─┼─┼─┼─┼─┼─┼─┼─┼─┼─┼─┼─┼ v2
1.5 ┼─┼─┼─┼─┼─┼─┼─┼─┼─┼─┼─┼─●
    ┼─┼─┼─┼─┼─┼─┼v3─┼─┼─┼─┼─┼
    ┼─┼─┼─┼─┼─┼─●─┼─┼─┼─┼─┼─┼
    ┼─┼─┼─┼─┼─┼─┼─┼─┼─┼─┼─┼─┼
    ┼v1─┼─┼─┼─┼─┼─┼─┼─┼─┼─┼─┼
    ●─┼─┼─┼─┼─┼─┼─┼─┼─┼─┼─┼─┼
   1           1.5          2

///////////////////////////////