- Distances
- Trigonometry
- Interpolation

A*A + B*B = C*C

Distance between two points P0 *(x0,y0)* and P1 *(x1,y1)*:

A = x1 - x0 B = y1 - y0 (x1-x0)*(x1-x0) + (y1-y0)*(y1-y0) = C*C dist = C = sqrt((x1-x0)*(x1-x0) + (y1-y0)*(y1-y0))

Distance between two 3D points P0 *(x0,y0,z0)* and P1 *(x1,y1,z1)*:

dist = sqrt((x1-x0)*(x1-x0) + (y1-y0)*(y1-y0) + (z1-z0)*(z1-z0))

*sqrt* is considered an expensive (i.e. slow) function

Avoid using it if you can

For example, to determine which of 2 points (P1 or P2) is closer to point P0:

dist1 = (x1-x0)*(x1-x0) + (y1-y0)*(y1-y0) + (z1-z0)*(z1-z0) dist2 = (x2-x0)*(x2-x0) + (y2-y0)*(y2-y0) + (z2-z0)*(z2-z0) if dist1 < dist2: P1 is closer else: P2 is closer

sin(A) = opposite / hypotenuse cos(A) = adjacent / hypotenuse tan(A) = opposite / adjacent

or

opposite = hypotenuse * sin(A) adjacent = hypotenuse * cos(a)

x = radius * cos(A) y = radius * sin(A)

Standard math library functions use radians

360 degrees = 1 full circle = 2 π radians

(Circumference of unit circle = 2 π)

radians = degrees / 360.0 * 2 * pi

or

radians = degrees / 180.0 * pi(or use Python's pre-defined functions

vertices = [] for degrees in range(0, 360): angleInRadians = math.radians(degrees) x = math.cos(angleInRadians) * radius y = math.sin(angleInRadians) * radius vertices += [x,y] vlist = pyglet.graphics.vertex_list(360, ('v2f', vertices))

Vehicle has direction and speed of travel

Direction is orientation - rotation about Z

To move forward:

distance = speed * time dx = math.cos(direction) * distance dy = math.sin(direction) * distance x = x + dx y = y + dy

`atan2` converts from (X,Y) coordinates back to angles

Note: it takes arguments in the order Y, X

angle = math.degrees( math.atan2(y,x) )

Deriving a value for something from two pre-defined values (extremes)

e.g. Moving object from one position to another, over time

Interpolation expressed as fractional distance between the two extremes

Ranges from 0.0 to 1.0

0.0 = first point; 1.0 = second point

For a single value, with extremes `V0` & `V1` and interpolation fraction `A`:

V = (1 - A) * V0 + A * V1

For multiple values, such as XYZ position, use the same fraction `A` for all:

X = (1 - A) * X0 + A * X1 Y = (1 - A) * Y0 + A * Y1 Z = (1 - A) * Z0 + A * Z1

To interpolate over time, compute interpolation fraction based on the amount of time that has passed.

Example:

def startAnimation(): animating = True startTime = time.time() duration = 5 def computeAnimation(): if animating: t = time.time() - startTime if t <= duration: a = t / duration else: animating = False a = 1 x = (1-a)*startX + a*endX y = (1-a)*startY + a*endY z = (1-a)*startZ + a*endZ

Linear | Slow-in Slow-out |
---|---|

X = t | X = -2*t*t*t + 3*t*t |

A2 = -2*A*A*A + 3*A*A V = (1-A2) * V0 + A2 * V1

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