matlab - 到 "curved"结构的欧氏距离

Question

我编写了一个程序，它处理当前资源到几何图形表面的欧氏距离。在我们的几何考虑中，y 坐标始终为零 - 因此它在 x 和 z 维度上是一个二维结构。该图显示了它的外观:

(length yellow = 100 , length blue = 500) -- 例如，如果原点位于第二个黄色部分的中间，则点源位于坐标 coord=(700,0,510)对于蓝色或黄色部分的每个中点，当然可以通过欧几里得距离:

平方(总和(((中点坐标))^2)

但是如果我从某个部分通过角度 $\alpha$ 来“弯曲”这个“棍子”，我该怎么做呢？:

y 坐标再次为零，源再次垂直放置在 z 方向上 - 但是我如何确定到截面中心的欧几里得距离？所以我们使用相同的长度尺寸，只是我们在电极的方向上将棒弯曲 30%，例如在原点或第 5 段。最好的方法是相应地操纵中点的“x”值....

Answer 1

我现在无法访问 Matlab，所以我用 python 编写了一些代码。这是你想做的吗？:

# plotting function of a polyline, a point and, if chosen, midpoints of polyline
def plot_polyline(P, P0, equal_axes, show_midpoints):
    M = midpoints_of(P)
    plt.figure()
    plt.plot(P0[0], P0[1], 'bo')
    plt.plot(P[:,0], P[:,1])
    for k in range(P.shape[0]):
        plt.plot(P[k,0], P[k,1], 'ro')
    if show_midpoints:
        for k in range(M.shape[0]):
            plt.plot(M[k,0], M[k,1], 'yo')
    axx = plt.gca()
    if equal_axes:
        axx.set_aspect('equal')
    plt.show()
    return None

def cos_sin(angle):
    return  math.cos(angle), math.sin(angle)

# generate a rotation matrix that rotates a 2D vector at an angle alpha
# observe the matrix format is [x1, y1] = [x, y].dot([[cos, -sin],
#                                                     [sin, cos]])
# and rotation is clockwise rotation of vector written as row-vectors 
def rot_matrx(angle):
    cs, sn = cos_sin(angle)
    return np.array([[  cs, -sn], 
                     [  sn,  cs]]) 

# performs a clockwise rotation of a point around a center at an angle 
def rotate(points, angle, center):
    U = rot_matrx(angle)
    return (points - center).dot(U) + center
 
# bending
def perform_bending(polyline, angle, index_bending_point):    
    x1 = polyline.copy()  
    x_bend  = x1[index_bending_point,:]
    # rotate half angle around point x_bend first         
    x1[0:index_bending_point,:] = rotate(x1[0:index_bending_point,:], 
                                               angle/2, x_bend) 
    # rotate the result from above half angle more around point x_(bend-1)   
    x_bend = x1[(index_bending_point-1),:]
    x1[0:(index_bending_point - 1),:] = rotate(x1[0:(index_bending_point - 1),:], 
                                                   angle/2, x_bend)      
    return x1

# calculate the array of midpoints of the polyline
def midpoints_of(polyline):
    k, m = polyline.shape
    return (polyline[0:(k-1), :] + polyline[1:k, :]) / 2

# calculate the distances from point to the midpoints of a polyline
def calc_distances(point, mids_of_polyline):
    return np.linalg.norm(point-mids_of_polyline, axis=1)


   
# initialize example:   
a = 2 # yellow segments' lenght
b = 5 # blue segments' lenght
n = 12 # number of segments + 1, number of points separating segments

alpha = 40  # angle of bending in degrees
alpha = math.pi * 40/180 # angle in bending in radians

# generate [0, 1, 2, 3, 4, 5, 6, 7]
I = np.arange(0,n)

# generate the x[0,] and x[1,] coordinates of the polygonal line
x = np.empty((n, 2), dtype=float)
I = np.floor(I / 2)*(a+b) + (I % 2)*a
x[:, 0] = I.T
x[:, 1] = 0

# point on x that is coordinate origin
x0 = (x[2,:] + x[3,:])/2
# shift initial polyline:
x = x - x0

# index of bending point
i_bend=5

x_bent = perform_bending(x, angle=alpha, index_bending_point=i_bend) 
x_mids = midpoints_of(x_bent)

# point
p = np.array([x_mids[i_bend-1, 0], 5.1])
    
dist = calc_distances(p, x_mids)

plot_polyline(x, p, equal_axes=True, show_midpoints=False)
plot_polyline(x_bent, p, equal_axes=True, show_midpoints=True)
plot_polyline(x_bent, p, equal_axes=True, show_midpoints=False)

print(dist)