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US-1889327: Fixed suggestions to matrix-representation-of-transformations.md and types-of-coordinate-systems.md file (#1265)
* Fixed suggestions for US-1889327 * Updated alt texts for US-1889327
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Pooja Poojari
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---
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title: "Matrix Representation of Transformations"
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ms.date: "03/30/2017"
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description: Learn about matrix representation of linear and composite transformations.
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ms.date: "01/06/2022"
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ms.custom: devdivchpfy22
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dev_langs:
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- "csharp"
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- "vb"
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@@ -19,13 +21,13 @@ ms.assetid: 0659fe00-9e0c-41c4-9118-016f2404c905
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---
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# Matrix Representation of Transformations
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An m×n matrix is a set of numbers arranged in m rows and n columns. The following illustration shows several matrices.
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:::image type="content" source="media/aboutgdip05-art04.gif" alt-text="Illustration of matrices.":::
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You can add two matrices of the same size by adding individual elements. The following illustration shows two examples of matrix addition.
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:::image type="content" source="media/aboutgdip05-art05.gif" alt-text="Illustration of matrix addition.":::
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An m×n matrix can be multiplied by an n×p matrix, and the result is an m×p matrix. The number of columns in the first matrix must be the same as the number of rows in the second matrix. For example, a 4×2 matrix can be multiplied by a 2×3 matrix to produce a 4×3 matrix.
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Points in the plane and rows and columns of a matrix can be thought of as vectors. For example, (2, 5) is a vector with two components, and (3, 7, 1) is a vector with three components. The dot product of two vectors is defined as follows:
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The following illustration shows several examples of matrix multiplication.
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:::image type="content" source="media/aboutgdip05-art06.gif" alt-text="Illustration of matrix multiplication.":::
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If you think of a point in a plane as a 1×2 matrix, you can transform that point by multiplying it by a 2×2 matrix. The following illustration shows several transformations applied to the point (2, 1).
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:::image type="content" source="media/aboutgdip05-art07.gif" alt-text="Matrix transformation to a point in a plane.":::
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All of the transformations shown in the preceding figure are linear transformations. Certain other transformations, such as translation, are not linear, and cannot be expressed as multiplication by a 2×2 matrix. Suppose you want to start with the point (2, 1), rotate it 90 degrees, translate it 3 units in the x direction, and translate it 4 units in the y direction. You can accomplish this by using a matrix multiplication followed by a matrix addition.
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:::image type="content" source="media/aboutgdip05-art08.gif" alt-text="Illustration of matrix multiplication followed by a matrix addition.":::
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A linear transformation (multiplication by a 2×2 matrix) followed by a translation (addition of a 1×2 matrix) is called an affine transformation. An alternative to storing an affine transformation in a pair of matrices (one for the linear part and one for the translation) is to store the entire transformation in a 3×3 matrix. To make this work, a point in the plane must be stored in a 1×3 matrix with a dummy 3rd coordinate. The usual technique is to make all 3rd coordinates equal to 1. For example, the point (2, 1) is represented by the matrix [2 1 1]. The following illustration shows an affine transformation (rotate 90 degrees; translate 3 units in the x direction, 4 units in the y direction) expressed as multiplication by a single 3×3 matrix.
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:::image type="content" source="media/aboutgdip05-art09.gif" alt-text="Illustration of an affine transformation.":::
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In the preceding example, the point (2, 1) is mapped to the point (2, 6). Note that the third column of the 3×3 matrix contains the numbers 0, 0, 1. This will always be the case for the 3×3 matrix of an affine transformation. The important numbers are the six numbers in columns 1 and 2. The upper-left 2×2 portion of the matrix represents the linear part of the transformation, and the first two entries in the 3rd row represent the translation.
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:::image type="content" source="media/aboutgdip05-art10.gif" alt-text="Illustration of linear and translation part of a matrix transformation.":::
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In GDI+ you can store an affine transformation in a <xref:System.Drawing.Drawing2D.Matrix> object. Because the third column of a matrix that represents an affine transformation is always (0, 0, 1), you specify only the six numbers in the first two columns when you construct a <xref:System.Drawing.Drawing2D.Matrix> object. The statement `Matrix myMatrix = new Matrix(0, 1, -1, 0, 3, 4)` constructs the matrix shown in the preceding figure.
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## Composite Transformations
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A composite transformation is a sequence of transformations, one followed by the other. Consider the matrices and transformations in the following list:
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|-|-|
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| Matrix | Transformation |
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|----------|----------------|
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|Matrix A|Rotate 90 degrees|
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|Matrix B|Scale by a factor of 2 in the x direction|
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|Matrix C|Translate 3 units in the y direction|
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@@ -80,8 +82,8 @@ An m×n matrix is a set of numbers arranged in m rows and n columns. The followi
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[2 1 1]D = [-2 5 1]
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The following illustration shows the matrices A, B, C, and D.
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:::image type="content" source="media/aboutgdip05-art12.gif" alt-text="Illustration of matrix A, B, C, and D.":::
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The fact that the matrix of a composite transformation can be formed by multiplying the individual transformation matrices means that any sequence of affine transformations can be stored in a single <xref:System.Drawing.Drawing2D.Matrix> object.
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@@ -94,8 +96,8 @@ An m×n matrix is a set of numbers arranged in m rows and n columns. The followi
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[!code-vb[System.Drawing.CoordinateSystems#11](~/samples/snippets/visualbasic/VS_Snippets_Winforms/System.Drawing.CoordinateSystems/VB/Class1.vb#11)]
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The following illustration shows the matrix.
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:::image type="content" source="media/aboutgdip05-art13.gif" alt-text="Matrix illustration of a composite transformation.":::
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## See also
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