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SSE SIMDを用いて４ｘ４の逆行列の高速アルゴリズム
Table of Contents トランスフォーム行列 一般の逆行列 付録その一 付録その二 English Verison 始まる前に、実際に必要となる逆行列は「一般の行列」かどうかを考えてください。 私は自作ゲームエンジンの数学系ライブラリを書く時、逆行列の問題を考えました。ゲームや３Dアプリケーションでは、オブジェクトのトランスフォーム情報は４ｘ４行列で記録されています。このような、位置、回転、スケールの三要素から作成している行列は、この文章で「トランスフォーム行列」と呼びます。トランスフォーム行列は一般の行列より2倍早い逆行列の求め方があります。この文章の前半は先ずトランスフォーム行列について話しましょう。後半はSIMD命令を用いた一般の４ｘ４行列の逆行列の求め方を説明します。最…
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Stop Using Normal Matrix
For rendering, I used to calculate normal matrix to transform vertex normal from model space to world space or view space. The normal matrix is defined as the inverse transpose of upper-left 3x3 matrix of the model matrix, from this article. Of course matrix inverse is not a cheap operation…
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Fast 4x4 Matrix Inverse with SSE SIMD, Explained
Table of Contents Transform Matrix Inverse General Matrix Inverse Appendix 1 Appendix 2 Before we start, think about this question: do we really need the inverse of a general matrix? I came to this problem when writing a math library for my game engine. If you are making a game…
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Conversion between View Space Linear and Screen Space Linear
Some background of this problem: when doing screen space reflection, we need to shoot a ray from every pixel to find out reflection color. However we don’t want to march the ray in view space, because for example you have a ray pointing relatively inwards or outwards to the…
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Calculate Minimal Bounding Sphere of Frustum
I came across this problem when fixing shadow shimmering in Cascaded Shadow Map, but it could be used in many other cases. To describe what we are going to do more specifically: given a frustum with near plane \(n\), far plane \(f\), and field of view angle \(fov\), we need…
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A Different Way to Understand Quaternion and Rotation
Before We Start Quaternion is widely used in game engines to represent 3D rotation. As a game engineer you might be using quaternion explicitly or implicitly in your daily work, but do you really understand what is going on under the hood when you are calling “rotate a vector” or…