Geometry - Scaling

Card Puncher Data Processing


Scaling is a transformation that is generally applied by the transformation matrix

See also: Linear Algebra - Scalar (Multiplication|Product) - Scaling

Matrix multiplication

The functional form <MATH> x' = a.x \\ y' = d.y </MATH> becomes the following matrix. <MATH> \begin{bmatrix} x' \\ y' \\ \end{bmatrix} = \begin{bmatrix} a & 0 \\ 0 & d \\ \end{bmatrix} \begin{bmatrix} x \\ y \\ \end{bmatrix} </MATH>

Using the standard transformation matrix notation, it would become: <MATH> \begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} = \begin{bmatrix} a & 0 & 0 \\ 0 & d & 0 \\ 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} </MATH>


One point

  • Scale Matrix Definition: To scale the point by two in the vertical direction, the corresponding matrix will be in two dimensions:

<MATH> \begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix} </MATH>

<MATH> \begin{bmatrix}x' \\ y'\end{bmatrix} = \begin{bmatrix}1x \\ 2y\end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix} \begin{bmatrix}x \\ y\end{bmatrix} </MATH>

Many points

To apply such a transformation to many points at the same time, the matrix-vector definition of matrix-matrix multiplication is used. The points are putted together to form a position matrix that is left-multiplied by the matrix representing the transformation: <MATH> \begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix} \begin{bmatrix} \begin{array}{r|r|r} x_1 & x_2 & x_3 \\ y_1 & y_2 & y_3 \end{array} \end{bmatrix} \begin{bmatrix} \begin{array}{r|r|r} 1x_1 & 1x_2 & 1x_3 \\ 2y_1 & 2y_2 & 2y_3 \end{array} \end{bmatrix} </MATH>

Recommended Pages
Card Puncher Data Processing
Geometry - Transformation

Transformation of geometric objects. A geometric transformation is represented by a transformation matrix. Transformations that keep the origin fixed are linear including: , ...
Matrix Rotation 90
Geometry - Transformation Matrix

A geometric transformation can be represented by a matrix. THE advantage of using transformation matrices is that cumulative transformations can be described by simply multiplying the matrices that describe...
Data System Architecture
Image/Aspect Ratio (width/heigth)

Aspect Ratio is the ratio between the width and the height of an image and has several impacts.
Inkscape Layout
Inkscape - Scale

in Inkscape Scale tab of the Transform dialog (Object → Transform... (Shift+Ctrl+M)). With the handle of the bounding box, see When an object(s) is selected < and >: Default scales...
Vector Arrow Multiplication
Linear Algebra - Scalar (Multiplication|Product) - Scaling

Scalar Multiplication (Scaling) is the multiplication of a vector (for instance ) by a scalar (real number) (for instance ) to produce another vector (for instance ) Multiplying a vector v by a scalar...
Map Projection
Spatial - Projection

See Projections: summarizing mutli-dimensional data in two or three dimensions. Data Projections Scale projection function Data domain Pixel Range DateTime (Years, Month, Date, Time) Pixel...
Ridge Regression Lambda Versus Standardized Coefficients
Statistics - Ridge regression

Ridge regression is a shrinkage method. It was invented in the '70s. The least squares fitting procedure estimates the regression parameters using the values that minimize RSS. In contrast, the...
Area Graph Sunspot 1700 2003
Viz - Graph Aspect Ratio

The aspect ratio of a graph between the height and width of the axis. rate of change0963488414The Elements of Graphing Data (1994) Viewers can best understand changes in the slope of the line when...
Utah Teapot
Viz - Key Frame (Target Value)

Key frames are images that defined the states of an animation. The images created between this key image are created with a process called inbetween or tweening. Keyframes are specific frames in the...

Share this page:
Follow us:
Task Runner