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computer graphics  

2017-10-09 20:20:51|  分类: 线性代数 |  标签: |举报 |字号 订阅

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The Geometry of 2 x 2 Matrices

Since a 2 x 2 matrix corresponds uniquely to a linear transformation from R2 to R2, we can think of a matrix as transforming a planar figure into a new planar figure.  

Example

Consider the matrix

        computer graphics - jia_huiqiang - 平安甜橙的博客

and the triangle with vertices (0,0), (1,2), (5,3).  We have

 

computer graphics - jia_huiqiang - 平安甜橙的博客

It is a property of linear transformations that if the matrix is nonsingular, then line segments map onto line segments.  Hence triangles map onto triangles.  The picture below shows the original triangle.

        computer graphics - jia_huiqiang - 平安甜橙的博客

Some Basic Transformations

There are certain basic transformation that are building blocks for general transformations.

Example  Reflection With Respect to the x axis.

To find the matrix for this transformation, we consider where the vectors e1 and e2 are mapped.  The reflection with respect to the x-axis makes the y-coordinate negative and leaves the x-coordinate constant.  We have

        L(1, 0)  =  (1, 0)        L(0, 1)  =  (0, -1)

These vectors are the column vectors for the matrix.  We have

        computer graphics - jia_huiqiang - 平安甜橙的博客

Example  Reflection About the Line y = x

We see that 

        L(1,0)  =  (0,1)        L(0,1)  =  (1,0)

so that 

        computer graphics - jia_huiqiang - 平安甜橙的博客

Example  Rotation About an Angle q

The point (0,1) rotated about this angle is on the unit circle at radian angle q.  The point (1,0) rotated about this angle is on the unit circle at radian angle p/2 + q.  We have

        L(1,0)  =  (cos q, sin q)        L(0,1)  =  (cos(p/2 + q), sin(p/2 + q)  =  (-sin q, cos q)

We have 

        computer graphics - jia_huiqiang - 平安甜橙的博客

Example  Shear in the y-direction

Another transformation that is common in computer graphics is a shear in the x or y direction.  The picture below gives and example

The matrix that makes this happen is 

        computer graphics - jia_huiqiang - 平安甜橙的博客

for some constant k

You can find an interactive applets that lets you play with computer graphics and matrices at

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