计算球面的旋转轴

Question

我在Maya中从头开始构建Sphere ，而不是使用球体顶点列表创建面，而是需要制作一个平面并将其旋转以使其与常规球体面匹配。

我的想法是获取球面的顶点在水平和垂直方向上的中心角。 这适用于Y轴，但是一旦我进行X旋转，脸部的方向就会丢失。

在图像中，我故意在X轴上旋转了一个球面，以说明需要计算哪种旋转。 该实现是用Python编写的，因此如果需要，我可以访问所有矢量方法。 请注意，此领域实现还有其他用途，因此设置似乎有些奇怪！

import pymel.core as pm
import pymel.core.datatypes as dt
import pymel.util as util

degrees = util.arrays.degrees
cos     = util.arrays.cos
sin     = util.arrays.sin
atan2   = util.math.atan2
acos    = util.math.acos
sqrt    = util.math.sqrt
PI      = util.arrays.pi
TWO_PI  = PI * 2

def distance(x1, y1, z1, x2, y2, z2):
    return sqrt( (x2 - x1) ** 2 + (y2 - y1) ** 2 + (z2 - z1) ** 2 )


# Sphere class
class Sphere():

# Initialise radius (float), subdivisionsAxis (int), subdivisionsHeight (int) 
def __init__(self, radius = 10, subdivisionsAxis = 8, subdivisionsHeight = 8):

    # Loop through each subdivision on y axis
    for i in range(subdivisionsHeight):

        if i == 0 or i == subdivisionsHeight - 1:

            # Store the triangle vertices's in this list
            data = self.generateSphereData(radius, subdivisionsAxis, subdivisionsHeight, i, 'triangle')

            length = len(data) / 11
            for j in range(length):
                index = j * 11
                x1 = data[index]
                y1 = data[index + 1]
                z1 = data[index + 2]
                x2 = data[index + 3]
                y2 = data[index + 4]
                z2 = data[index + 5]
                x3 = data[index + 6]
                y3 = data[index + 7]
                z3 = data[index + 8]
                # Angle y
                ay = data[index + 9]
                # Angle z
                az = data[index + 10]

                v1 = dt.FloatVector(x1, y1, z1)
                v2 = dt.FloatVector(x2, y2, z2)
                v3 = dt.FloatVector(x3, y3, z3)

                # Ignore the top and bottom triangles for now...
                # pm.polyCreateFacet( p = [ v1, v2, v3 ] )

            else:
                # Store the quads vertices's in this list
                data = self.generateSphereData(radius, subdivisionsAxis, subdivisionsHeight, i, 'quad')

                length = len(data) / 14

                for j in range(length):
                    index = j * 14
                    x1 = data[index]
                    y1 = data[index + 1]
                    z1 = data[index + 2]
                    x2 = data[index + 3]
                    y2 = data[index + 4]
                    z2 = data[index + 5]
                    x3 = data[index + 6]
                    y3 = data[index + 7]
                    z3 = data[index + 8]
                    x4 = data[index + 9]
                    y4 = data[index + 10]
                    z4 = data[index + 11]
                    # Angle y
                    ay = data[index + 12]
                    # Angle z
                    az = data[index + 13]

                    v1 = dt.FloatVector(x1, y1, z1)
                    v2 = dt.FloatVector(x2, y2, z2)
                    v3 = dt.FloatVector(x3, y3, z3)
                    v4 = dt.FloatVector(x4, y4, z4)

                    # Calculate centroid
                    cx = (x1 + x2 + x3 + x4) / 4
                    cy = (y1 + y2 + y3 + y4) / 4
                    cz = (z1 + z2 + z3 + z4) / 4

                    # Calculate the width and height

                    # Calculate dimensions for facet
                    tw = distance(x1, y1, z1, x4, y4, z4)
                    bw = distance(x2, y2, z2, x3, y3, z3)
                    w  = tw if bw < tw else bw
                    h  = distance(x2, y2, z2, x1, y1, z1)

                    # Calculate rotation of face
                    centroid = dt.FloatVector(cx, cy, cz)

                    mesh = pm.polyPlane(width=1, height=1, subdivisionsX=1, subdivisionsY=1, axis=(1, 0, 0))
                    mesh[0].setTranslation(centroid)
                    mesh[0].setRotation([0, degrees(-ay), 0])

                    pm.spaceLocator(p=v1)
                    pm.spaceLocator(p=v2)
                    pm.spaceLocator(p=v3)
                    pm.spaceLocator(p=v4)

                    # pm.polyCreateFacet( p = [ v1, v2, v3, v4 ] )


# Generate a vertex list of the spheres current subdivision height level
# Arguments: radius (float), subdivisionsAxis (int), subdivisionsHeight (int), index (int), polygonType (string) 

def generateSphereData(self, radius, subdivisionsAxis, subdivisionsHeight, index, polygonType):
    positions = []

    if polygonType == 'triangle':
        for i in range(subdivisionsAxis):

            # If were generating the top triangles we need the triangle base to 
            # Be at the previous subdivision level, so change the index to index - 1
            if index < subdivisionsHeight: 
                nextIndex = index + 1
            else:                
                nextIndex = index - 1

            if i < subdivisionsAxis - 1:
                j = i + 1
            else:
                j = 0

            # Top vertex
            r1 = radius  * sin(index * (PI / subdivisionsAxis))
            x1 = r1      * cos(i * (TWO_PI / subdivisionsAxis))
            y1 = radius  * cos(index * (PI / subdivisionsHeight))
            z1 = r1      * sin(i * (TWO_PI / subdivisionsAxis))

            # Left vertex
            r2 = radius  * sin(nextIndex * (PI / subdivisionsAxis))
            x2 = r2      * cos(i * (TWO_PI / subdivisionsAxis))
            y2 = radius  * cos(nextIndex * (PI / subdivisionsHeight))
            z2 = r2      * sin(i * (TWO_PI / subdivisionsAxis))

            # Right vertex
            x3 = r2      * cos(j * (TWO_PI / subdivisionsAxis))
            y3 = radius  * cos(nextIndex * (PI / subdivisionsHeight))
            z3 = r2      * sin(j * (TWO_PI / subdivisionsAxis))

            # Calculate angles
            ay = 0
            az = 0

            positions += [x1, y1, z1, x2, y2, z2, x3, y3, z3, ay, az]

    elif polygonType == 'quad':

        nextIndex = index + 1

        for i in range(subdivisionsAxis):

            if i < subdivisionsAxis - 1:
                j = i + 1
            else:
                j = 0

            # Bottom y
            r1 = radius * sin(index * (PI / subdivisionsAxis))
            y1 = radius * cos(index * (PI / subdivisionsHeight))

            # Top y
            r2 = radius * sin(nextIndex * (PI / subdivisionsAxis))
            y2 = radius * cos(nextIndex * (PI / subdivisionsHeight))

            # Top left vertex
            x1 = r2     * cos(i * (TWO_PI / subdivisionsAxis))
            z1 = r2     * sin(i * (TWO_PI / subdivisionsAxis))

            # Bottom left vertex
            x2 = r1     * cos(i * (TWO_PI / subdivisionsAxis))
            z2 = r1     * sin(i * (TWO_PI / subdivisionsAxis))

            # Bottom right vertex
            x3 = r1     * cos(j * (TWO_PI / subdivisionsAxis))
            z3 = r1     * sin(j * (TWO_PI / subdivisionsAxis))

            # Top right vertex
            x4 = r2     * cos(j * (TWO_PI / subdivisionsAxis))
            z4 = r2     * sin(j * (TWO_PI / subdivisionsAxis))

            # Calculate angles
            ay1 = i * (TWO_PI / subdivisionsAxis)
            ay2 = j * (TWO_PI / subdivisionsAxis)
            ay  = ay1 + ((ay2 - ay1) / 2)

            az1 = index     * (PI / subdivisionsHeight)
            az2 = nextIndex * (PI / subdivisionsHeight)
            az  = az1 + ((az2 - az1) / 2)

            positions += [x1, y2, z1, x2, y1, z2, x3, y1, z3, x4, y2, z4, ay, az]

    return positions

Sphere(20, 8, 8)

Answer 1

好的，伪代码。 这个怎么样：

planeNormal = cross(plane.firstEdge, plane.secondEdge)
faceNormal = cross(face.firstEdge, face.secondEdge)

normalize(planeNormal)
normalize(faceNormal)

if dot(planeNormal, faceNormal)<0    # if they're more than 90 degrees apart
  rotate(plane, plane.firstEdge, pi) # rotate the plane 180 degrees
  planeNormal = -planeNormal

axis = cross(planeNormal, faceNormal)
angle = arccos(magnitude(axis))
normalize(axis)
rotate(plane, axis, angle)

Answer 2

如果我正确阅读了您的问题，则希望对一组四边形和三角形进行旋转，以使它们成为一个球面。 我相信您真正要问的是如何旋转四边形和三边形的集合，以使垂直于形状中心的向量与所需点相交。 所需的点是球体的中心。

让我们分手吧！ 这是您要执行的操作的伪代码：

for i,shape in enumerate(listOfShapes):
    [nc1,nc2] = normal_of_center(shape)
    R = rotationFrom2Vecs([nc1,nc2],[s,nc2])
    listOfShapes[i] = shape*R

对于每个形状（形状只是3点或4点的列表），您将计算一个垂直于该形状并以该形状的中心为中心的向量。 该向量仅是两个点：[nc1，nc2]（提示nc2：只是所讨论形状的平均坐标，nc1是位于该形状定义的平面法线上的任何点）。

我们要做的下一件事是计算此向量与将形状中心（nc2）连接到球体中心的向量之间的旋转矩阵R。 但是，您可能会问自己，如何从2个向量计算3x3旋转矩阵？

最后，您可以通过右乘法将此旋转矩阵应用于形状，由于矩阵乘法的规则，这是唯一有效的方法。 我对pymel不熟悉，不知道它是否具有矩阵乘法例程。 如果不是这样，您将希望使用numpy数组和矩阵，这些数组和矩阵对于此类项目非常简单并且方便。

如果您对整体想法感到困惑，我可以绘制一些快速而肮脏的图表。 这是最快的方法吗？ 否。但这绝对是最清晰的方法。