The general matrix form

A geometric transformation maps each point of an object to a corresponding point in its image. Write the original point as a coordinate column, then transform it using

Apply the matrix first, then add the translation.

For a shape, apply the rule to each vertex separately. Plot the image vertices in the corresponding order before reconnecting them. Depending on the matrix, the transformation may alter position, orientation, size or shape.

Triangle $ABC$ under five matrix transforms.

Reflections

A reflection is a rigid transformation. It maps every point to the opposite side of a mirror line, keeping the same perpendicular distance from that line. Common reflections about lines through the origin are

For instance, reflecting in $y=x$ exchanges the two coordinates. If the mirror line doesn’t pass through the origin, translations are required before and after the reflection.

Stretches and enlargements

A horizontal stretch is a non-rigid transformation that multiplies every horizontal coordinate by a scale factor. Vertical coordinates stay unchanged:

In the same way, a vertical stretch multiplies every vertical coordinate by a scale factor but leaves the horizontal coordinate unchanged:

Scale factors between $0$ and $1$ produce a compression. A negative factor reverses the relevant direction as well, combining scaling with reflection.

An enlargement about the origin is a similarity transformation that multiplies every distance from the origin by one common scale factor:

Unequal horizontal and vertical stretches can change shape and angles; an enlargement preserves both.

Rotations

A rotation is a rigid transformation that turns each point through the same angle around a fixed centre. For an anticlockwise rotation about the origin, the matrix is

When $\theta$ is negative, the rotation is clockwise. To rotate about another centre, translate that centre to the origin, perform the rotation, then translate back.

Translations

A translation is a rigid transformation that moves every point by the same displacement without changing its size or orientation. It can be written as

Two-dimensional matrix multiplication alone cannot perform a translation because a matrix always sends the origin to the origin. The added column in the general form is therefore necessary.

Applications and technology

These rules are examples of affine transformations: mappings formed from a linear matrix transformation followed by a translation. In digital image processing, they act on pixel coordinates to rotate, resize, reflect or reposition an image. If transformed coordinates don’t fall exactly on pixel centres, interpolation is needed.

Computer graphics uses matrices to position three-dimensional models and convert them into two-dimensional views. Related techniques in healthcare help reconstruct, align and visualize medical scans. By allowing clinicians to compare scans and inspect structures from useful viewpoints, this can advance diagnosis. The quality of the conclusion, however, still depends on the source data and the assumptions in the model.

Combining transformations

A composition of transformations is one mapping created by applying two or more transformations in a specified order. The order matters. If one transformation comes first and another follows, the matrix for the second transformation multiplies on the left.

Suppose the first rule is $\boldsymbol{X}'=\boldsymbol{A}\boldsymbol{X}+\boldsymbol{t}$ and the second rule is

Substituting the first rule into the second gives

The combined matrix is therefore $\boldsymbol{B}\boldsymbol{A}$, which is not generally the same as $\boldsymbol{A}\boldsymbol{B}$. Matrix multiplication is not commutative, so reversing the order may produce a different image. In a longer composition, begin with the transformation applied first and write the matrix order explicitly.

Iteration and fractals

Iteration is a repeated process where the output of one step becomes an input for the next. Repeated affine transformations can be expressed as

A fractal is a geometric structure produced by a repeated rule, with detail or similar patterns appearing across multiple scales. In a triangular gasket, for example, several contractions move points towards different corners. A snowflake-like boundary can be made in another way: repeatedly replace each segment with a fixed arrangement of shorter segments. Both methods are algorithms. Their forms arise from precise repeated instructions, rather than from drawing more and more intricate details by hand.

If several transformations are available, transformation $i$ may be chosen with probability $p_i$. Here, $p_i$ is its dimensionless selection probability, and the probabilities sum to $1$. Altering these probabilities changes how densely different regions are visited, even if the transformations remain fixed. This provides a mathematical analogy for biological mutation models: changing the likelihood of different rules can affect which features appear frequently in the generated form.

The probabilistic process also links to Markov chains. When the probability of choosing the next rule depends only on the current rule or state, a transition matrix can describe the choices. When each selection is independent of all earlier choices, fixed probabilities are enough, making the process simpler than a general Markov chain.

Repeated contractions connect with infinite geometric series as well. If a displacement is repeatedly scaled by a dimensionless common ratio $r$, its total scale is

Because the series converges, infinitely many movements that become successively smaller can approach a finite limiting position.

A nonlinear fractal recurrence

Some fractal algorithms are not affine matrix iterations. One well-known quadratic complex recurrence is

One possible starting point is $Z_0=c$, where $Z_0$ is the initial complex value. Changing the initial value or $c$ can determine whether the sequence remains bounded, altering the resulting fractal boundary. The contrast is useful: affine iterations repeatedly transform points geometrically, while this recurrence repeatedly applies a nonlinear transformation to a complex number.

What counts as an explanation?

The question “Do mathematicians and historians mean the same thing by explain?” is particularly relevant here. A mathematician might explain a fractal by stating an algorithm and proving that its consequences follow from the rule. A historian might instead explain why people developed or adopted that algorithm by referring to their aims and circumstances. Both accounts offer explanations, but they address different questions. One focuses on logical mechanism; the other focuses on context and causation.

Determinant as an area scale factor

The determinant of a two-dimensional transformation matrix is a scalar found from the matrix entries. It gives the signed area scale factor of the transformation. For the general matrix,

Areas are related by

where $S_{\text{image}}$ is the area of the image (square coordinate units), and $S_{\text{object}}$ is the area of the original object (square coordinate units). Since area cannot be negative, the absolute value is essential.

Translation isn’t included in this formula: moving a figure leaves its area unchanged. So the full affine rule $\boldsymbol{A}\boldsymbol{X}+\boldsymbol{t}$ has the same area scale factor as the matrix part $\boldsymbol{A}$.

The determinant’s magnitude and sign tell us different things:

• If $|\det(\boldsymbol{A})|>1$, area increases.
• If $0<|\det(\boldsymbol{A})|<1$, area decreases.
• If $|\det(\boldsymbol{A})|=1$, area is preserved.
• If $\det(\boldsymbol{A})<0$, orientation is reversed as well as area being scaled.
• If $\det(\boldsymbol{A})=0$, every two-dimensional region collapses onto a line or a point, so the image has zero area and the transformation cannot be reversed.

For a horizontal stretch followed by a vertical stretch, the determinant is $k_hk_v$. Geometrically, one dimension is multiplied by $k_h$ and the other by $k_v$, so the area is multiplied by $|k_hk_v|$. An enlargement with scale factor $k$ has area scale factor $k^2$. Reflections have determinant $-1$; rotations have determinant $1$.

Determinants of compositions

For two successive matrix transformations,

There is a direct geometric interpretation. The first transformation scales area by $|\det(\boldsymbol{A})|$. The second then scales that area by $|\det(\boldsymbol{B})|$, making the overall area factor the product of the two factors. Here, an algebraic matrix property describes a visible geometric effect.