Question

我正在尝试使用Newell的方法来计算Python中的表面法线向量，基于here中的以下伪代码。

Begin Function CalculateSurfaceNormal (Input Polygon) Returns Vector

   Set Vertex Normal to (0, 0, 0)

   Begin Cycle for Index in [0, Polygon.vertexNumber)

      Set Vertex Current to Polygon.verts[Index]
      Set Vertex Next    to Polygon.verts[(Index plus 1) mod Polygon.vertexNumber]

      Set Normal.x to Sum of Normal.x and (multiply (Current.y minus Next.y) by (Current.z plus Next.z))
      Set Normal.y to Sum of Normal.y and (multiply (Current.z minus Next.z) by (Current.x plus Next.x))
      Set Normal.z to Sum of Normal.z and (multiply (Current.x minus Next.x) by (Current.y plus Next.y))

   End Cycle

   Returning Normalize(Normal)

End Function

这是我的代码：

Point3D = collections.namedtuple('Point3D', 'x y z')

def surface_normal(poly):
    n = [0.0, 0.0, 0.0]

    for i, v_curr in enumerate(poly):
        v_next = poly[(i+1) % len(poly)]
        n[0] += (v_curr.y - v_next.y) * (v_curr.z - v_next.z)
        n[1] += (v_curr.z - v_next.z) * (v_curr.x - v_next.x)
        n[2] += (v_curr.x - v_next.x) * (v_curr.y - v_next.y)

    normalised = [i/sum(n) for i in n]

    return normalised

def test_surface_normal():
    poly = [Point3D(0.0, 0.0, 0.0),
            Point3D(0.0, 1.0, 0.0),
            Point3D(1.0, 1.0, 0.0),
            Point3D(1.0, 0.0, 0.0)]

    assert surface_normal(poly) == [0.0, 0.0, 1.0]

这在标准化步骤失败，因为此时n为[0.0, 0.0, 0.0]。如果我理解正确，那应该是[0.0, 0.0, 1.0]（Wolfram Alpha的confirmed）。

我在这里做错了什么？有没有更好的方法来计算python中的表面法线？我的多边形总是平面的，所以如果有另一种方式，Newell的方法并不是绝对必要的。

Answer 1

如果您想要Newell方法的替代方法，您可以使用2个非平行向量的交叉乘积。这应该适用于您提供的任何平面形状。我知道该理论认为它适用于凸多边形，但我们在Wolfram Alpha上看到的示例为偶数凹多边形（例如bowtie多边形）返回了合适的曲面法线。

Answer 2

好的，问题实际上是一个愚蠢的问题。

如下的行：

n[0] += (v_curr.y - v_next.y) * (v_curr.z - v_next.z)

应该是：

n[0] += (v_curr.y - v_next.y) * (v_curr.z + v_next.z)

应添加第二组括号中的值，而不是相减。

Answer 3

我认为有必要澄清有关Newell方法的信息性实现的几点。初读https://stackoverflow.com/users/1706564/jamie-bull更新时，我认为符号的变化仅涉及第一个等式。但实际上是所有这些。其次，提供跨产品方法的比较。最后，处理除以零的问题-但可以除以epsilon。我也用numpy代替。

import numpy as np

def surface_normal_newell(poly):

    n = np.array([0.0, 0.0, 0.0])

    for i, v_curr in enumerate(poly):
        v_next = poly[(i+1) % len(poly),:]
        n[0] += (v_curr[1] - v_next[1]) * (v_curr[2] + v_next[2]) 
        n[1] += (v_curr[2] - v_next[2]) * (v_curr[0] + v_next[0])
        n[2] += (v_curr[0] - v_next[0]) * (v_curr[1] + v_next[1])

    norm = np.linalg.norm(n)
    if norm==0:
        raise ValueError('zero norm')
    else:
        normalised = n/norm

    return normalised


def surface_normal_cross(poly):

    n = np.cross(poly[1,:]-poly[0,:],poly[2,:]-poly[0,:])

    norm = np.linalg.norm(n)
    if norm==0:
        raise ValueError('zero norm')
    else:
        normalised = n/norm

    return normalised


def test_surface_normal3():
    """ should return:

        Newell:
        Traceback (most recent call last):
          File "demo_newells_surface_normals.py", line 96, in <module>
            test_surface_normal3()
          File "demo_newells_surface_normals.py", line 58, in test_surface_normal3
            print "Newell:", surface_normal_newell(poly) 
          File "demo_newells_surface_normals.py", line 24, in surface_normal_newell
            raise ValueError('zero norm')
        ValueError: zero norm
    """
    poly = np.array([[1.0,0.0,0.0],
                     [1.0,0.0,0.0],
                     [1.0,0.0,0.0]])
    print "Newell:", surface_normal_newell(poly) 


def test_surface_normal2():
    """ should return:

        Newell: [ 0.08466675 -0.97366764 -0.21166688]
        Cross : [ 0.08466675 -0.97366764 -0.21166688]
    """
    poly = np.array([[6.0,1.0,4.0],
                     [7.0,0.0,9.0],
                     [1.0,1.0,2.0]])
    print "Newell:", surface_normal_newell(poly)
    print "Cross :", surface_normal_cross(poly)


def test_surface_normal1():
    """ should return:

        Newell: [ 0.  0. -1.]
        Cross : [ 0.  0. -1.]
    """
    poly = np.array([[0.0,1.0,0.0],
                     [1.0,1.0,0.0],
                     [1.0,0.0,0.0]])
    print "Newell:", surface_normal_newell(poly) 
    print "Cross :", surface_normal_cross(poly)


print "Test 1:"
test_surface_normal1()
print "\n"

print "Test 2:"
test_surface_normal2()
print "\n"

print "Test 3:"
test_surface_normal3()

使用Newell方法

3 个答案: