Algebra for imaging
The matrices that transform: products, rotation, scaling and shear matrices, homogeneous coordinates for translation, composing in the right order, inverses and change of frame.
- 30 cards
- 5 topics
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Each topic is weighted the way the exam weighs it.
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A few cards from the deck
Matrix product: what is it?
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A matrix whose every entry is the dot product of a row of the first matrix with a column of the second.
It is not commutative: A·B and B·A generally differ. That is why the order of transformations matters.
R is the 90° rotation [[0, −1], [1, 0]] and S the scaling [[2, 0], [0, 1]]. What is R · S?
[[0, −1], [2, 0]][[0, −2], [1, 0]][[0, 0], [0, 0]][[2, −1], [1, 1]]
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[[0, −1], [2, 0]]Row 1 of R by the columns of S: (0·2 + (−1)·0, 0·0 + (−1)·1) = (0, −1). Row 2: (1·2 + 0·0, 1·0 + 0·1) = (2, 0).
[[0, −2], [1, 0]]is S · R, the opposite order; the zero matrix multiplies entry by entry;[[2, −1], [1, 1]]adds.The product A·B exists when the number of … of A equals the number of … of B.
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The product A·B exists when the number of columns of A equals the number of rows of B.
Each entry combines a row of A with a column of B, which must have the same length. A 2 × 3 matrix times a 3 × 4 matrix gives a 2 × 4 matrix.
Transformation matrix: what is it?
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A matrix whose product with a point's coordinates gives the coordinates of its image under a transformation.
Its columns are the images of the basis vectors: that of the x axis, then that of the y axis. Reading them is enough to recognise the transformation.
An object must spin on itself, then be placed at (10, 0). The code computes R · T · p and the object travels around a circle about the world origin. Which fix?
- Replace T with its inverse before the product
- Write T · R · p: the rotation must act before the translation
- Transpose R so that it turns the other way
- Switch from 3 × 3 matrices to 2 × 2 matrices
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Write T · R · p: the rotation must act before the translation
In R · T · p, T first moves the point to (10, 0), then R rotates it about the world origin: it orbits. In T · R · p, the rotation happens about the object's own origin, then it is moved.