154 lines
6.8 KiB
Markdown
154 lines
6.8 KiB
Markdown
# orx-math
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Mathematical utilities, including complex numbers,
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linear ranges, simplex ranges, matrices and radial basis functions (RBF).
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<!-- __demos__ -->
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## Demos
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### linearrange/DemoLinearRange02
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Demonstrate how to create a 1D linear range between two instances of a `LinearType`, in this case,
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a horizontal `Rectangle` and a vertical one.
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Notice how the `..` operator is used to construct the `LinearRange1D`.
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The resulting `LinearRange1D` provides a `value()` method that takes a normalized
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input and returns an interpolated value between the two input elements.
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This example draws a grid of rectangles interpolated between the horizontal and the vertical
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triangles. The x and y coordinates and the `seconds` variable are used to specify the
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interpolation value for each grid cell.
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One can use the `LinearRange` class to construct
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- a `LinearRange2D` out of two `LinearRange1D`
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- a `LinearRange3D` out of two `LinearRange2D`
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- a `LinearRange4D` out of two `LinearRange3D`
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(not demonstrated here)
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[source code](src/jvmDemo/kotlin/linearrange/DemoLinearRange02.kt)
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### linearrange/DemoLinearRange03
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Demonstrates how to create a `LinearRange2D` out of two `LinearRange1D` instances.
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The first range interpolates a horizontal rectangle into a vertical one.
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The second range interpolates two smaller squares of equal size, one placed
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higher along the y-axis and another one lower.
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A grid of such rectangles is displayed, animating the `u` and `v` parameters based on
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`seconds`, `x` and `y` indices. The second range results in a vertical wave effect.
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[source code](src/jvmDemo/kotlin/linearrange/DemoLinearRange03.kt)
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### matrix/DemoLeastSquares01
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Demonstrate least squares method to find a regression line through noisy points.
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The line drawn in red is the estimated line. The green one is the ground-truth.
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`Ax = b => x = A⁻¹b`
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because `A` is likely inconsistent, we look for an approximate `x` based on `AᵀA`, which is consistent.
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`x̂ = (AᵀA)⁻¹ Aᵀb`
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[source code](src/jvmDemo/kotlin/matrix/DemoLeastSquares01.kt)
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### matrix/DemoLeastSquares02
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Demonstrate how to use the `least squares` method to fit a cubic bezier to noisy points.
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On every animation frame, 10 concentric circles are created centered on the window and converted to contours.
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In OPENRNDR, circular contours are made ouf of 4 cubic-Bezier curves. Each of those curves is considered
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one by one as the ground truth, then 5 points are sampled near those curves.
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Finally, two matrices are constructed using those points and math operations are applied to
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revert the randomization attempting to reconstruct the original curves.
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The result is drawn on every animation frame, revealing concentric circles that are more or less similar
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to the ground truth depending on the random values used.
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[source code](src/jvmDemo/kotlin/matrix/DemoLeastSquares02.kt)
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### rbf/RbfInterpolation01
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Demonstrates using a two-dimensional Radial Basis Function (RBF) interpolator
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with the user provided 2D input points, their corresponding values (colors in this demo),
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a smoothing factor, and a radial basis function kernel.
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The program chooses 14 random points in the window area leaving a 100 pixels
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margin around the borders and assigns a randomized color to each point.
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Next it creates the interpolator using those points and colors, a smoothing factor
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and the RBF function used for interpolation. This function takes a squared distance
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as input and returns a scalar value representing the influence of points at that distance.
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A ShadeStyle implementing the RBF interpolation is created next, used to render
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the background gradient interpolating all points and their colors.
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After rendering the background, the original points and their colors are
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drawn as circles for reference.
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Finally, the current mouse position is used for sampling a color
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from the interpolator and displayed for comparison. Notice that even if
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the fill color is flat, it may look like a gradient due to the changing
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colors in the surrounding pixels.
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[source code](src/jvmDemo/kotlin/rbf/RbfInterpolation01.kt)
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### rbf/RbfInterpolation02
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Demonstrates using a two-dimensional Radial Basis Function (RBF) interpolator
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with the user provided 2D input points, their corresponding values (colors in this demo),
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a smoothing factor, and a radial basis function kernel.
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The program chooses 20 random points in the window area leaving a 100 pixels
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margin around the borders and assigns a randomized color to each point.
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Next it creates the interpolator using those points and colors, a smoothing factor
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and the RBF function used for interpolation. This function takes a squared distance
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as input and returns a scalar value representing the influence of points at that distance.
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A ShadeStyle implementing the same RBF interpolation is created next, used to render
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the background gradient interpolating all points and their colors.
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After rendering the background, the original points and their colors are
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drawn as circles for reference.
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Finally, the current mouse position is used for sampling a color
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from the interpolator and displayed for comparison. Notice that even if
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the fill color is flat, it may look like a gradient due to the changing
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colors in the surrounding pixels.
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[source code](src/jvmDemo/kotlin/rbf/RbfInterpolation02.kt)
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### simplexrange/DemoSimplexRange3D01
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Demonstrates the use of the `SimplexRange3D` class. Its constructor takes 4 instances of a `LinearType`
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(something that can be interpolated linearly, like `ColorRGBa`). The `SimplexRange3D` instance provides
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a `value()` method that returns a `LinearType` interpolated across the 4 constructor arguments using
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a normalized 3D coordinate.
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This demo program creates a 3D grid of 20x20x20 unit 3D cubes. Their color is set by interpolating
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their XYZ index across the 4 input colors.
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2D, 4D and ND varieties are also provided by `SimplexRange`.
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Simplex Range* is not to be confused with *Simplex Noise*.
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[source code](src/jvmDemo/kotlin/simplexrange/DemoSimplexRange3D01.kt)
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