在JavaScript中乘以4x3仿射变换矩阵 [英] Multiplying 4x3 affine transform matrices in javascript
问题描述
我正在尝试在Three.JS中计算转换，说实话，我是3D数学的新手.
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.
我有一组输入和正确的输出.我通过使用python来获得它，因为有一个名为Noesis的库，该库具有4x3矩阵的乘法.
问题在于该库是从python二进制文件中调用的，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，我希望能够使用javascript做到这一点.
问题:
- 有可用的图书馆吗?
- 什么是公式?
python模块 Noesis 具有以下内容: :
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noesis.mat43Mul的源代码.
更新
正在测试 Tamas Hegedus 答案:
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);
但是此modelSpaceMat的结果与python库的modelSpaceMat有点不同.就像位置部分在python中是(-1.1457012760729413e-07, 30.441999435424805, 9.592490357590577e-08)但在javascript中是3.863013744354248, 30.815059661865234, -7.705572128295898.
那是为什么?
作为参考，以下是console.log的输出:
从数学的角度来看，您不能将4x3矩阵相乘.
当涉及仿射变换时，图形库在齐次空间中使用特殊的平方矩阵子集来描述该变换:
a b c 0
d e f 0
g h i 0
x y z 1
最后一列被选择为0 0 0 1，因此变换不是透视投影.
某些图形库仅存储该转换的有趣的4x3部分，但是必须像4x4形式那样进行乘法.
Threejs具有内置的 Matrix4 类，该类实现乘法. /p>
I am trying to calculate transforms in Three.JS and honestly I'm new to 3D math.
In Three.JS there is a 3x3 and 4x4 matrix class but no 4x3.
Goal: I want to understand how to do multiplication on a 4x3*4x3 matrix.
I have a set of the inputs and the correct outputs. I got it by using python, because there is a library called Noesis that has multiplication of 4x3 matrices.
The problem is that the library calls from a python binary, so I can't see the source code.
The code in python is simply: modelSpaceMat = boneMat * frameMat
Here is a sample of the correct dataset:
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)
)
Now I want to be able to do it with javascript.
Questions:
- Is there any library available?
- What is the formula?
The python module, Noesis, has this line:
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])
I can't find the source code for noesis.mat43Mul.
UPDATE
Testing-out Tamas Hegedus answer:
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);
But the result of this modelSpaceMat is a bit different to modelSpaceMat of the python library. Like the position part which is (-1.1457012760729413e-07, 30.441999435424805, 9.592490357590577e-08) in python but is 3.863013744354248, 30.815059661865234, -7.705572128295898 in javascript.
Why is that?
For reference here are the outputs of the console.log:
From a mathematical point of view, you cannot multiply 4x3 matrices.
When its about affine transformations, graphics libraries use a special subset of square matrices in the homogeneous space to describe the transformation:
a b c 0
d e f 0
g h i 0
x y z 1
The last column is chosen to be 0 0 0 1 so the transformation is not a perspective projection.
Some graphics libraries only store the interesting 4x3 portion of that transformation, but multiplication must be done as it was in the 4x4 form.
Threejs has a builtin Matrix4 class, which implements multiplication.
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