The Need for Homogeneous Coordinates
In standard Cartesian coordinates, a point in 2D space is represented by two values (x, y), and in 3D space by three values (x, y, z). These representations work well for most operations but have limitations, particularly when trying to represent translations (movements in space) as matrix operations.
The Role of 0s and 1 in Homogeneous Coordinates
To overcome these limitations and uniformly apply all affine transformations (including translations) using matrix multiplication, we introduce an additional dimension, often denoted as w, leading to homogeneous coordinates.
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For Points: By setting w = 1, we enable translations.
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Here’s why: When you multiply a transformation matrix by a point in homogeneous coordinates , the 1 at the end allows the translation values from the matrix to be added to the point’s x, y, and z coordinates.
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This is because the matrix’s translation components are effectively multiplied by 1 (the w value for points) and then added to the point’s coordinates, enabling the point to move in space.
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For Vectors: Setting w = 0 for vectors means translations do not apply.
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Vectors represent direction and magnitude, not positions. When w = 0, the translation components of the matrix do not affect the vector because multiplying them by 0 nullifies their effect.
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This is appropriate since translating a direction or magnitude without altering its nature doesn’t conceptually make sense.
Mathematical Consistency
The inclusion of 0s and 1 in the matrix, specifically in its last row and column, ensures the matrix can handle both points and vectors appropriately within the same mathematical framework. The 0s in the last row ensure that when the matrix is used to transform vectors, the translation components do not contribute to the result. The 1 in the bottom-right corner ensures that the matrix multiplication is mathematically valid and maintains the dimensionality of the space.
Practical Example
Consider a simple translation of a point (2, 3) by (5, -1) in 2D:
Without homogeneous coordinates, we’d just add the translation to the point: (2 + 5, 3 - 1) = (7, 2).
With homogeneous coordinates, the point becomes (2, 3, 1), and the translation is represented by a matrix:
Multiplying this matrix by the point, we get the same result, (7, 2, 1), but through a consistent matrix operation that can also include rotation, scaling, etc., in the same step.
Thus, the 0s and 1 in homogeneous coordinates and transformation matrices enable a unified, consistent approach to applying various transformations, including translations, to both points and vectors within the same mathematical operations.