在javascript中乘以4x3仿射变换矩阵

Question

我正在尝试计算Three.JS中的变换,老实说,我是3D数学的新手.

在Three.JS中有3x3和4x4矩阵类,但没有4x3.

目标:我想了解如何在4x3*4x3矩阵上进行乘法运算.

我有一组输入和正确的输出.我通过使用python得到它,因为有一个名为Noesis4x3矩阵的乘法库.

问题是库调用python二进制文件,所以我看不到源代码.

python中的代码很简单: modelSpaceMat = boneMat * frameMat

以下是正确数据集的示例:

boneMat= (
 (0.0, 0.0, 1.0), 
 (0.0, 1.0, 0.0), 
 (-1.0, 0.0, 0.0), 
 (0.0, 32.29199981689453, -3.2665998935699463)
)
frameMat= (
 (0.6425124406814575, -0.06018795818090439, 0.7639083862304688), 
 (-0.003379624802619219, 0.9966778755187988, 0.08136376738548279), 
 (-0.7662678360939026, -0.05486864596605301, 0.640174925327301), 
 (4.438972473144531, -1.4769394397735596, 3.863013744354248)
)
modelSpaceMat=(
 (-0.7639083862304688, -0.06018795818090439, 0.6425124406814575), 
 (-0.08136376738548279, 0.9966778755187988, -0.003379624802619219), 
 (-0.640174925327301, -0.05486864596605301, -0.7662678360939026), 
 (-1.1457012760729413e-07, 30.441999435424805, 9.592490357590577e-08)
)

boneMat= (
 (1.0, 0.0, 0.0), 
 (0.0, 1.0, 0.0), 
 (0.0, 0.0, 1.0), 
 (0.0, 0.007000000216066837, -3.2665998935699463)
)
frameMat= (
 (1.0000004768371582, 2.3320662876358256e-06, -1.4419805665966123e-05), 
 (2.817355607476202e-06, 0.9999991059303284, -3.3202520626218757e-06), 
 (1.5038006495160516e-05, -2.133270072590676e-06, 1.0000003576278687), 
 (-0.0015083501348271966, 0.02300064079463482, 3.266691207885742)
)
modelSpaceMat= (
 (1.0000004768371582, 2.3320662876358256e-06, -1.4419805665966123e-05), 
 (2.817355607476202e-06, 0.9999991059303284, -3.3202520626218757e-06), 
 (1.5038006495160516e-05, -2.133270072590676e-06, 1.0000003576278687), 
 (-0.0014612301019951701, 0.030011480674147606, 9.013115777634084e-05)
)

现在我希望能够用javascript做到这一点.

问题:

• 有没有图书馆？
• 公式是什么？

---

python模块Noesis有这一行:

def __mul__(self, other):
    if isinstance(other, (NoeMat43, list, tuple)):
        return noesis.mat43Mul(self, other)
    elif isinstance(other, NoeVec3):
        return noesis.mat43TransformPoint(self, other)
    elif isinstance(other, NoeVec4):
        return noesis.mat43TransformVec4(self, other)
    else:
        return NoeMat43([self.mat43[0]*other, self.mat43[1]*other, self.mat43[2]*other, self.mat43[3]*other])

我找不到源代码noesis.mat43Mul.

---

UPDATE

测试Tamas Hegedus回答:

var boneMat = new THREE.Matrix4().copy(_bone.userData.modelMatrix);

var frameMat = new THREE.Matrix4();
frameMat.set(
  bone.rotationMatrix[0], bone.rotationMatrix[1], bone.rotationMatrix[2], bone.positionVector[0],
  bone.rotationMatrix[3], bone.rotationMatrix[4], bone.rotationMatrix[5], bone.positionVector[1],
  bone.rotationMatrix[6], bone.rotationMatrix[7], bone.rotationMatrix[8], bone.positionVector[2],
  0.0,     0.0,     0.0,     1.0
);

var modelSpaceMat = new THREE.Matrix4();
modelSpaceMat.multiplyMatrices(boneMat, frameMatrix);

console.log(boneMat, frameMat, modelSpaceMat);

但结果与python库modelSpaceMat有点不同modelSpaceMat.就像(-1.1457012760729413e-07, 30.441999435424805, 9.592490357590577e-08)在python中的位置部分,但是3.863013744354248, 30.815059661865234, -7.705572128295898在javascript中.

这是为什么？

这里参考的是console.log:

Answer 1

从数学的角度来看,你不能乘以4x3矩阵.

当它关于仿射变换时,图形库在同构空间中使用特殊的矩形矩阵子集来描述变换:

a b c 0
d e f 0
g h i 0
x y z 1

选择最后一列是为了0 0 0 1使变换不是透视投影.

有些图形库只存储该变换的有趣4x3部分,但乘法必须按照4x4格式进行.

Threejs有一个内置Matrix4类,它实现了乘法.