3D Transformations

Every rotation, stretch and mirror in a game is a 3×3 matrix. Its columns say where the x, y and z axes land, and the whole model follows.

where

\hat\imath

landswhere

\hat\jmath

landswhere

\hat k

landseigenvectorsvolume × 1.00

How it works

Linear maps, determinants and eigenvectors

In one sentence

A linear map sends each vector $\mathbf v$ to $A\mathbf v$ : the columns of $A$ are the images of the basis vectors, $\det A$ is the factor by which volumes scale, and eigenvectors are the directions $A$ only stretches.

General formula

{\htmlData{sym=0}{\mathbf{v}'}} = {\htmlData{sym=2}{A}}\,{\htmlData{sym=1}{\mathbf{v}}}, \qquad {\htmlData{sym=3}{\det A}} = \text{volume scale}

Watch where the three axis arrows land. That is all of $A$ ; every other point follows by linearity.

What the symbols mean

$\mathbf{v}'$
Transformed point
Where the point ends up after the matrix acts on it.
$\mathbf{v}$
Original point
A point, or an arrow from the origin, before the transformation.
$A$
Transformation matrix
Its columns are where the three axis arrows land. Everything else follows along.
$\det A$
Determinant
How much A scales volume. Negative means it flips space inside out; zero means it flattens it.

The columns of $A$ are the images of $\hat\imath$ , $\hat\jmath$ and $\hat k$ . Every other point is a combination of those three.
$\det A$ is how much volume grows. Negative means the model was mirrored, zero means it was flattened.
An eigenvector only stretches: $A\mathbf{v} = \lambda\mathbf{v}$ . It marks a direction the transformation leaves pointing the same way.

Learn the math

Vectors: The BasicsCalc III · Lesson 1, Vectors Change of VariablesCalc III · Lesson 37, Multiple Integrals Linear TransformationsLin Alg · Lesson 7, Linear Equations in Linear Algebra The Matrix of a Linear TransformationLin Alg · Lesson 8, Linear Equations in Linear Algebra Matrix OperationsLin Alg · Lesson 9, Matrix Algebra Matrices in Computer GraphicsLin Alg · Lesson 15, Matrix Algebra Introduction to DeterminantsLin Alg · Lesson 16, Determinants Change of BasisLin Alg · Lesson 24, Vector Spaces Eigenvectors and EigenvaluesLin Alg · Lesson 25, Eigenvalues and Eigenvectors Eigenvectors and Linear TransformationsLin Alg · Lesson 28, Eigenvalues and Eigenvectors Eigenvalues and EigenvectorsDiff Eq · Lesson 31, Systems of Differential Equations

Learn the math

Vectors: The BasicsCalc III · Lesson 1, Vectors

Change of VariablesCalc III · Lesson 37, Multiple Integrals

Linear TransformationsLin Alg · Lesson 7, Linear Equations in Linear Algebra

The Matrix of a Linear TransformationLin Alg · Lesson 8, Linear Equations in Linear Algebra

Matrix OperationsLin Alg · Lesson 9, Matrix Algebra

Matrices in Computer GraphicsLin Alg · Lesson 15, Matrix Algebra

Introduction to DeterminantsLin Alg · Lesson 16, Determinants

Change of BasisLin Alg · Lesson 24, Vector Spaces

Eigenvectors and EigenvaluesLin Alg · Lesson 25, Eigenvalues and Eigenvectors

Eigenvectors and Linear TransformationsLin Alg · Lesson 28, Eigenvalues and Eigenvectors

Eigenvalues and EigenvectorsDiff Eq · Lesson 31, Systems of Differential Equations