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https://github.com/PixelGuys/Cubyz.git
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Optimize and simplify the CaveBiomeMap value interpolation, making use of the axis aligned nature of the underlying grid.
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IntegratedQuantum
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@ -201,38 +201,54 @@ pub const InterpolatableCaveBiomeMapView = struct { // MARK: InterpolatableCaveB
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fn rotate312(in: Vec3i) Vec3i {
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return @shuffle(i32, in, undefined, Vec3i{2, 0, 1});
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}
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fn argMaxDistance0(distance: Vec3i) @Vector(3, bool) {
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fn argMaxDistance0(distance: Vec3i) u2 {
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const absDistance = @abs(distance);
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if(absDistance[0] > absDistance[1]) {
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if(absDistance[0] > absDistance[2]) {
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return .{true, false, false};
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return 0;
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} else {
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return .{false, false, true};
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return 2;
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}
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} else {
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if(absDistance[1] > absDistance[2]) {
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return .{false, true, false};
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return 1;
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} else {
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return .{false, false, true};
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return 2;
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}
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}
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}
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fn argMaxDistance1(distance: Vec3i) @Vector(3, bool) {
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fn argMaxDistance1(distance: Vec3i) u2 {
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const absDistance = @abs(distance);
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if(absDistance[0] >= absDistance[1]) {
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if(absDistance[0] >= absDistance[2]) {
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return .{true, false, false};
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return 0;
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} else {
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return .{false, false, true};
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return 2;
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}
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} else {
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if(absDistance[1] >= absDistance[2]) {
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return .{false, true, false};
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return 1;
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} else {
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return .{false, false, true};
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return 2;
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}
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}
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}
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fn vectorElement(val: Vec3i, i: u2) i32 {
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return switch(i) {
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0 => val[0],
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1 => val[1],
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2 => val[2],
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else => unreachable,
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};
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}
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fn indexToBool(i: u2) @Vector(3, bool) {
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return switch(i) {
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0 => .{true, false, false},
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1 => .{false, true, false},
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2 => .{false, false, true},
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else => unreachable,
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};
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}
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/// Return either +1 or -1 depending on the sign of the input number.
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fn nonZeroSign(in: Vec3i) Vec3i {
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@ -261,92 +277,29 @@ pub const InterpolatableCaveBiomeMapView = struct { // MARK: InterpolatableCaveB
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const worldPos = CaveBiomeMapFragment.rotate(.{wx, wy, wz});
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const closestGridpoint0 = (worldPos +% @as(Vec3i, @splat(CaveBiomeMapFragment.caveBiomeSize/2))) & @as(Vec3i, @splat(~@as(i32, CaveBiomeMapFragment.caveBiomeMask)));
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const distance0 = worldPos -% closestGridpoint0;
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const step0 = @select(i32, argMaxDistance0(distance0), @as(Vec3i, @splat(CaveBiomeMapFragment.caveBiomeSize)), @as(Vec3i, @splat(0)));
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const coordinate0 = argMaxDistance0(distance0);
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const step0 = @select(i32, indexToBool(coordinate0), @as(Vec3i, @splat(CaveBiomeMapFragment.caveBiomeSize)), @as(Vec3i, @splat(0)));
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const secondGridPoint0 = closestGridpoint0 +% step0*nonZeroSign(distance0);
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const closestGridpoint1 = (worldPos & @as(Vec3i, @splat(~@as(i32, CaveBiomeMapFragment.caveBiomeMask)))) +% @as(Vec3i, @splat(CaveBiomeMapFragment.caveBiomeSize/2));
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const distance1 = worldPos -% closestGridpoint1;
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const step1 = @select(i32, argMaxDistance1(distance1), @as(Vec3i, @splat(CaveBiomeMapFragment.caveBiomeSize)), @as(Vec3i, @splat(0)));
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const coordinate1 = argMaxDistance1(distance1);
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const step1 = @select(i32, indexToBool(coordinate1), @as(Vec3i, @splat(CaveBiomeMapFragment.caveBiomeSize)), @as(Vec3i, @splat(0)));
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const secondGridPoint1 = closestGridpoint1 +% step1*nonZeroSign(distance1);
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const @"r⃗₄" = closestGridpoint0;
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const @"r⃗₃" = secondGridPoint0;
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const @"r⃗₂" = closestGridpoint1;
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const @"r⃗₁" = secondGridPoint1;
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// Doing tetrahedral interpolation between the given points.
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// Barycentric coordinates for tetrahedra:
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// λ₄ = 1 - λ₁ - λ₂ - λ₃
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// ┌ ┐ ┌ ┐ ┌ ┐
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// | x₁-x₄ x₂-x₄ x₃-x₄ | |λ₁| | x-x₄ |
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// | y₁-y₄ y₂-y₄ y₃-y₄ | · |λ₂| = | y-y₄ | =: d⃗
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// | z₁-z₄ z₂-z₄ z₃-z₄ | |λ₃| | z-z₄ |
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// └ ┘ └ ┘ └ ┘
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// \_________ __________/
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// \/
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// =: A
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const @"d⃗" = distance0;
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const @"d⃗₁" = @"r⃗₁" -% @"r⃗₄";
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const @"d⃗₂" = @"r⃗₂" -% @"r⃗₄";
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const @"d⃗₃" = @"r⃗₃" -% @"r⃗₄";
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// With some renamings we get:
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// ┌ ┐
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// A = | d⃗₁ d⃗₂ d⃗₃ |
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// └ ┘
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const coordinateFinal = 3 ^ coordinate0 ^ coordinate1;
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const interpFinal = @abs(0.5 + @as(f32, @floatFromInt(vectorElement(distance0, coordinateFinal))))*2/CaveBiomeMapFragment.caveBiomeSize;
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// The inverse of a 3×3 matrix is given by:
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// ┌ ┐
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// | a₁₁ a₁₂ a₁₃ |
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// A = | a₂₁ a₂₂ a₂₃ |
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// | a₃₁ a₃₂ a₃₃ |
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// └ ┘
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// ┌ ┐
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// | |a₂₂ a₂₃| |a₁₃ a₁₂| |a₁₂ a₁₃| |
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// | |a₃₂ a₃₃| |a₃₃ a₃₂| |a₂₂ a₂₃| |
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// 1 | |
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// A⁻¹ = ––– | |a₂₃ a₂₁| |a₁₁ a₁₃| |a₁₃ a₁₁| |
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// |A| | |a₃₃ a₃₁| |a₃₁ a₃₃| |a₂₃ a₂₁| |
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// | |
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// | |a₂₁ a₂₂| |a₁₂ a₁₁| |a₁₁ a₁₂| |
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// | |a₃₁ a₃₂| |a₃₂ a₃₁| |a₂₁ a₂₂| |
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// └ ┘
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// Resolving the determinants gives:
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// ┌ ┐
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// | a₂₂·a₃₃ - a₂₃·a₃₂ a₁₃·a₃₂ - a₁₂·a₃₃ a₁₂·a₂₃ - a₁₃·a₂₂ |
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// 1 | |
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// A⁻¹ = ––– | a₂₃·a₃₁ - a₂₁·a₃₃ a₁₁·a₃₃ - a₁₃·a₃₁ a₁₃·a₂₁ - a₁₁·a₂₃ |
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// |A| | |
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// | a₂₁·a₃₂ - a₂₂·a₃₁ a₁₂·a₃₁ - a₁₁·a₃₂ a₁₁·a₂₂ - a₁₂·a₂₁ |
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// └ ┘
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// Notice how each column represents a rotated row of the original matrix.
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const row1 = Vec3i{@"d⃗₁"[0], @"d⃗₂"[0], @"d⃗₃"[0]};
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const row2 = Vec3i{@"d⃗₁"[1], @"d⃗₂"[1], @"d⃗₃"[1]};
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const row3 = Vec3i{@"d⃗₁"[2], @"d⃗₂"[2], @"d⃗₃"[2]};
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const determinantCol1 = rotate231(row2)*rotate312(row3) - rotate312(row2)*rotate231(row3);
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const determinantCol2 = rotate312(row1)*rotate231(row3) - rotate231(row1)*rotate312(row3);
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const determinantCol3 = rotate231(row1)*rotate312(row2) - rotate312(row1)*rotate231(row2);
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// Notice that the determinant |A| can be expressed as dot(row1, determinantCol1)
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const determinantA = vec.dot(determinantCol1, row1);
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const invDeterminantA = 1.0/@as(f32, @floatFromInt(determinantA));
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// Now we change the memory layout use rows instead of columns to make matrix-vector multiplication easier later.
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const determinantRow1 = Vec3i{determinantCol1[0], determinantCol2[0], determinantCol3[0]};
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const determinantRow2 = Vec3i{determinantCol1[1], determinantCol2[1], determinantCol3[1]};
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const determinantRow3 = Vec3i{determinantCol1[2], determinantCol2[2], determinantCol3[2]};
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const interp0 = 0.5 + (@abs(@as(f32, @floatFromInt(vectorElement(distance0, coordinate0))))/CaveBiomeMapFragment.caveBiomeSize - 0.5)/(1 - interpFinal);
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const interp1 = 0.5 + (@abs(@as(f32, @floatFromInt(vectorElement(distance1, coordinate1))))/CaveBiomeMapFragment.caveBiomeSize - 0.5)/interpFinal;
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const @"unscaledλ123" = Vec3i{vec.dot(determinantRow1, @"d⃗"), vec.dot(determinantRow2, @"d⃗"), vec.dot(determinantRow3, @"d⃗")};
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const @"λ1" = @as(f32, @floatFromInt(@"unscaledλ123"[0]))*invDeterminantA;
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const @"λ2" = @as(f32, @floatFromInt(@"unscaledλ123"[1]))*invDeterminantA;
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const @"λ3" = @as(f32, @floatFromInt(@"unscaledλ123"[2]))*invDeterminantA;
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const @"λ4" = 1 - @"λ1" - @"λ2" - @"λ3";
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// TODO: I wonder if there are some optimizations possible, given that
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// per construction |x₁ - x₄| = |x₂ - x₄| = ... = |z₂ - z₄| = ±caveBiomeSize/2
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// And |r⃗₃ - r⃗₄| = caveBiomeSize, where 2 elements are 0
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// Now after all this math we can finally do what we actually want: Interpolate the damn thing.
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const biome4 = self._getBiome(closestGridpoint0[0], closestGridpoint0[1], closestGridpoint0[2], 0);
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const biome3 = self._getBiome(secondGridPoint0[0], secondGridPoint0[1], secondGridPoint0[2], 0);
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const biome2 = self._getBiome(closestGridpoint1[0], closestGridpoint1[1], closestGridpoint1[2], 1);
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const biome1 = self._getBiome(secondGridPoint1[0], secondGridPoint1[1], secondGridPoint1[2], 1);
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return @field(biome1, field)*@"λ1" + @field(biome2, field)*@"λ2" + @field(biome3, field)*@"λ3" + @field(biome4, field)*@"λ4";
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const biome00 = self._getBiome(closestGridpoint0[0], closestGridpoint0[1], closestGridpoint0[2], 0);
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const biome01 = self._getBiome(secondGridPoint0[0], secondGridPoint0[1], secondGridPoint0[2], 0);
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const biome10 = self._getBiome(closestGridpoint1[0], closestGridpoint1[1], closestGridpoint1[2], 1);
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const biome11 = self._getBiome(secondGridPoint1[0], secondGridPoint1[1], secondGridPoint1[2], 1);
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const val0 = @field(biome00, field)*(1 - interp0) + @field(biome01, field)*interp0;
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const val1 = @field(biome10, field)*(1 - interp1) + @field(biome11, field)*interp1;
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return val0*(1 - interpFinal) + val1*interpFinal;
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}
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/// On failure returnHeight contains the lower border of the terrain height.
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