确定一个平面的角度.. [英] determine the angle of a plane..

问题描述

我在我的directX模型中创建下面的平面的矩阵代码如下所示：

 FrameTransformMatrix {
 1.0,0.0,0.0,0.0，
 0.0,1.0,0.0,0.0，
 0.0,0.0,1.0,0.0，
 2750.0,0.000163,0.000143,1.0 ;; 
} 

在C＃中，我可以从顶点缓冲区中的现有点创建一个平面。

 Vector3 [] vectors =  new  Vector3 [mesh.MeshParts [ 0 ] VertexBuffer.VertexCount]。 
 mesh.MeshParts [ 0 ]。VertexBuffer.GetData< Vector3>（向量）; 
 Plane plane =  new  Plane（vectors [mesh.MeshParts [ 0 ]。StartIndex]，向量[mesh.MeshParts [ 0 ]。StartIndex +  1 ]，向量[mesh.MeshParts [ 0 ]。StartIndex +  2 ]）; 

现在，有了这些信息，在我的渲染代码的effect.World部分，我如何渲染一个对象，所有3个轴对齐并接触平面？假设要放置的对象有一个以Vector3.Zero为中心的基础。



可以使用C＃和XNA中现有的Matrix函数完成任务，还是数学需要要在其他地方完成吗？

解决方案

给定给定点[p]处的平面向量[n]，并且你有一个绝对原点的对象模型，而你想要在与平面对齐的点[p]处渲染对象，那么你必须找到以下转换：

1. T ₁ =通过绝对z轴（即[n]与[n]投影到绝对xy平面的投影的[n]角度，将绝对y轴围绕绝对y轴旋转绝对y轴= acos（））。
2. T ₂ =将绝对xy平面绕绝对z轴旋转绝对xy平面中的[n] - 角（即角度）在从上面的[n]投影到绝对x轴的绝对xy平面中。
3. T ₃ =转换为点[p]


此转换序列将您的对象转换为平面的点[p]及其对齐的轴。

一旦找到这些转换T ₁ ，T ₂ ，T ₃ ，你将变换的T _x 乘以如下：

T =（T < sub> 1 ^T x T ₂ ^T x T ₃ ^T ） ^T



（M ^T 是M的转置矩阵，即镜像在对角线的左上角到右下角）



使用该T变换矩阵，您将每个对象的点相乘，从而获得在所需位置渲染的对象。

例如v _plane = T xv _origin



给定

[n] = [ n _x ，n _y ，n _z ，1] ^T

[p ] = [p _x ，p _y ，p _z ，1] ^T

q = sqrt（n _x ² + n _y ² ）

r = sqrt （n _x ² + n _y ² + n _z ² ）



T ₁ =

< td> 0

名词<子>ž / R，	0，	q / R，	0
0，	1，	0，	0
-q / r，	0，	n z / r，
0，	0，	0，	1





T ₂ =

< td> 0，

n x / q，	-n y q，	0，	0
n y / q，	ñ x / q，	0，	0
0，	0，	1，	0
0，	0，	1





T ₃ =

< td> 1，

1，	0，	0，	p x
0，	0，	p y
0，	0，	1，	p z
0，	0，	0，	1




我将T矩阵的计算作为练习;-)



干杯

Andi

The matrix code in my directX model from which I am creating the plane below looks like this;

FrameTransformMatrix {
  1.0, 0.0, 0.0, 0.0,
  0.0, 1.0, 0.0, 0.0,
  0.0, 0.0, 1.0, 0.0,
  2750.0, 0.000163, 0.000143, 1.0;;
}

In C# I can create a plane from existing points in the vertex buffer.

Vector3[] vectors = new Vector3[mesh.MeshParts[0].VertexBuffer.VertexCount];
mesh.MeshParts[0].VertexBuffer.GetData<Vector3>(vectors);
Plane plane = new Plane(vectors[mesh.MeshParts[0].StartIndex], vectors[mesh.MeshParts[0].StartIndex + 1], vectors[mesh.MeshParts[0].StartIndex + 2]);

Now, with this information, in the effect.World portion of my render code, how would I render an object with all 3 axes aligned to and touching the plane? Assume the object to be placed has a base centered at Vector3.Zero.

Can the task be accomplished with the existing Matrix functions in C# and XNA or will the math need to be done elsewhere?

解决方案

Given the plane vector [n] at a given point [p], and you have an object model at the absolute origin, and you want to render the object at point [p] aligned with the plane, then you must find the following transformation:

1. T₁ = rotate the absolute x-z plane around the absolute y-axis by the [n]-angle in the plane through absolute z-axis (i.e. angle of [n] to the projection of [n] to the absolute x-y plane = acos()).
2. T₂ = rotate the absolute x-y plane around the absolute z-axis by the [n]-angle in the absolute x-y plane (i.e. the angle in the absolute x-y plane from the [n] projection above to the absolute x-axis).
3. T₃ = translate to point [p]

This transformation sequence transforms your object to the plane's point [p] and its aligned axes.
Once you found these transformation T₁, T₂, T₃, you multiply the transformed T_x as follows:
T = (T₁^T x T₂^T x T₃^T)^T

(M^T is the transposed matrix of M, i.e. mirrored at the diagonal top-left to bottom-right)

With that T transformation matrix, you multiply each object's point and so get the object rendered at the desired location.
E.g. v_plane = T x v_origin

Given
[n] = [n_x, n_y, n_z, 1]^T
[p] = [p_x, p_y, p_z, 1]^T
q = sqrt(n_x² + n_y²)
r = sqrt(n_x² + n_y² + n_z²)

T₁ =

nz/r,	0,	q/r,	0
0,	1,	0,	0
-q/r,	0,	nz/r,	0
0,	0,	0,	1

T₂ =

nx/q,	-nyq,	0,	0
ny/q,	nx/q,	0,	0
0,	0,	1,	0
0,	0,	0,	1

T₃ =

1,	0,	0,	px
0,	1,	0,	py
0,	0,	1,	pz
0,	0,	0,	1

I leave the calculation of the T-matrix as exercise ;-)

Cheers
Andi

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