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Contemporary geometry concerns 2d and 3d structures, and is fundamentally important to understand. Geometry relies on some basic algebraic principles, which will also be covered here. For computer graphics, geometry can be used to define primitive shapes.

The most important types of numbers with respect to digital art are integers and decimal numbers (fractions):


integers on a number line, always being displayed as whole numbers with no decimal indication. they can be positive or negative with respect to 0

fractions and decimals

mathematical truths with fractions on left of the equals sign and decimals on right

All geometry is based on the cartesian coordinate system, which defines points in space based on axes and an origin located at the center:

cartesian plane

2d cartesian system


2d origin

cartesian xyz

3d cartesian system

cartesian point

general representation for point a

Several different 2d geometric shapes exist:

2d shapes

most common 2d shapes. a shape is a polygon, and can have any positive integer number of sides greater than 2

triangle area

triangle area is b*h/2

square area

square area is length*width

polygon area

polygon area is P*A/2

Several different 3d geometric solids exist:

3d solids

most common 3d solids


sphere area is 4*3.1415*r^2;

circular cylinder

cylinder area is 2*3.1415*radius*height


torus area is 4*3.1415^2*(pink radius)*(red radius)


cone area is 3.1415*r*(r^2+h^2)^.5

In order to understand more specific constructs, a more firm grasp of algebra is required. All numbers can be represented by variables, which can then have basic operations thereof including (but not limited to): adding, subtracting, multiplying, dividing, exponential and parenthesis all of which have applicable processing rules. The most important rule is the order of operations, which states the operations must be performed in this order from first to last:

  1. parenthesis
  2. exponential
  3. multiply
  4. divide
  5. add
  6. subtract

which may be remembered with the mnemonic "PEMDAS".

operation order

mathematical truth demonstrating MDAS from above, using integer substitution to start


algebraic truths

algebra and fractions

algebraic fractional truth

algebra and exponents

algebraic exponential truths

A function is a mathematical variable output set, each value for which is determined by a variable input:

parabolic function

function for a parabola

parabolic plot

plot of a parabola

another parabolic plot

other parabolic functions / plots

img src="img/learn3d/x2y2.gif" alt="paraboloid"/>

plot of a paraboloid

paraboloid function output

function outputs for the paraboloid equation; the numbers indicate a similarity to a parabola output

step function plot

plot of a step function

step function description

description of the step function