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What are the best practices for applying affine transformations to 2D and 3D objects?

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Affine transformations are mathematical operations that can change the shape, size, position, orientation, and perspective of 2D and 3D objects in computer graphics. They are useful for creating realistic animations, manipulating images, and applying effects. However, applying affine transformations correctly can be tricky, especially when dealing with multiple transformations and different coordinate systems. In this article, you will learn some of the best practices for applying affine transformations to 2D and 3D objects, such as using homogeneous coordinates, matrix multiplication, order of operations, and inverse transformations.

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Harish Chouhan

1 Homogeneous coordinates

One of the best practices for applying affine transformations to 2D and 3D objects is to use homogeneous coordinates. Homogeneous coordinates are a way of representing points and vectors in a higher-dimensional space, where an extra coordinate is added to indicate the scale or weight of the point. For example, a 2D point (x, y) can be represented as a 3D homogeneous coordinate (x, y, w), where w is not zero. This allows affine transformations to be represented as matrix operations, which makes them easier to perform and combine. For example, a translation by (a, b) can be represented as a 3x3 matrix that multiplies the homogeneous coordinate by adding a and b to the x and y coordinates.

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2 Matrix multiplication

Another best practice for applying affine transformations to 2D and 3D objects is to use matrix multiplication. Matrix multiplication is a way of combining multiple affine transformations into a single matrix that can be applied to the object. For example, if you want to rotate an object by an angle theta, scale it by a factor s, and then translate it by (a, b), you can multiply the matrices for each transformation in the reverse order, and then multiply the result by the object's homogeneous coordinates. Matrix multiplication is also useful for transforming objects between different coordinate systems, such as world, camera, and screen coordinates.

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Harish Chouhan
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An affine transform has 3 key parameters: alpha - scale theta - rotation delta - translation. In general an output can be expressed as [Xnew Ynew] = alpha * R [ Xold Yold ] + T , where Matrix R = [ cos(theta) -sin(theta) sin(theta) cos(theta) ] and translation vector T = [ Tx Ty] Affine transforms find may applications in real life. One important application is image compression. Identify a basic image tile and see what transformations of this tile build the entire image. Store only the basic tile and the transformations and not the original image.

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3 Order of operations

A third best practice for applying affine transformations to 2D and 3D objects is to pay attention to the order of operations. The order of operations determines how the object is transformed by multiple affine transformations, and can affect the final appearance and position of the object. For example, if you want to rotate an object around a point other than the origin, you need to first translate the object to the origin, then rotate it, and then translate it back to the original position. The order of operations also affects how the object is transformed between different coordinate systems, such as world, camera, and screen coordinates. The order of operations is usually determined by the matrix multiplication order, which is usually from right to left.

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Harish Chouhan
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Affine transformation are Linear Transformation in any space. For example the geometry that an object had before the transform will be retained after the transform. This is in contrast to non-linear transformation. Non-linear transformation will warp the geometry of the object post the transformation. Let us take an example of a line. A line will remain to be a line after the Affine transformation. If you use a non-linear transform, then the line will become a curve.

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4 Inverse transformations

A fourth best practice for applying affine transformations to 2D and 3D objects is to use inverse transformations. Inverse transformations are the opposite of affine transformations, and can undo or reverse the effect of an affine transformation. For example, if you want to undo a translation by (a, b), you can apply an inverse translation by (-a, -b). Inverse transformations are useful for restoring the original state of an object, or for transforming objects in the opposite direction. For example, if you want to transform a point from screen coordinates to world coordinates, you can apply the inverse of the transformation that converts a point from world coordinates to screen coordinates. Inverse transformations can be obtained by finding the inverse matrix of the affine transformation matrix, or by applying the inverse operations in the reverse order.

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5 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?

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What are the best practices for applying affine transformations to 2D and 3D objects?

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2

3

4

5

1 Homogeneous coordinates

2 Matrix multiplication

3 Order of operations

4 Inverse transformations

5 Here’s what else to consider

Computer Graphics

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What are the best practices for applying affine transformations to 2D and 3D objects?

1

2

3

4

5

1 Homogeneous coordinates

2 Matrix multiplication

3 Order of operations

4 Inverse transformations

5 Here’s what else to consider

Computer Graphics

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