数学图形之将曲线(curve)转化成曲面管

在我关于数学图形的博客中,一开始讲曲线的生成算法.然后在最近的章节中介绍了圆环,还介绍了螺旋管以及海螺的生成算法.一类是曲线,一类是环面,为什么不将曲线变成环的图形,毕竟曲线看上去太单薄了,这一节我将介绍如何依照曲线(Curve)生成其相应的曲面管.

相关软件参见:数学图形可视化工具,使用自己定义语法的脚本代码生成数学图形.

在圆柱面这一节的最后,我提供了两种算法,以生成朝向任意方向的圆柱面,一种是极坐标原理,另一种是矩阵原理.这一节也是采用这两个原理生成曲面管,因为由曲线生成的曲面管就是若干个有朝向的圆柱面组成.

先看个三维曲线的代码:

圆环面螺线:

#http://xuxzmail.blog.163.com/blog/static/25131916200976114621705/
#Toroidal spiral

vertices = 

t = from  to (*PI)

r =
n = 

x = (r+sin(*t))*cos(t)
y = (r+sin(*t))*sin(t)
z = cos(n*t)

r = ;
x = x*r
y = y*r
z = z*r

圆环面螺线管(极坐标原理):

#http://xuxzmail.blog.163.com/blog/static/25131916200976114621705/
#Toroidal spiral

vertices = D1: D2:

u = from  to (*PI) D1
v = from  to (*PI) D2

r =
n = 

xq = (r+sin(*v))*cos(v)
yq = (r+sin(*v))*sin(v)
zq = cos(n*v)

#由线变管

xd = array_difference(xq, )
yd = array_difference(yq, )
zd = array_difference(zq, )

len = sqrt(xd*xd + yd*yd + zd*zd)
a = xd/len
b = yd/len
c = zd/len

radius =0.2

ac = sqrt(a*a + c*c)
angleY = atan2(ac, b)
angleXZ = atan2(a,c)

m = cos(u) * radius
n = sin(u) * radius

e = sin(angleY)
f = cos(angleY)

g = sin(angleXZ)
h = cos(angleXZ)

x0 = m
y0 = -n*e
z0 = n*f

x1 = x0*h + z0*g
y1 = y0
z1 = -x0*g + z0*h

x = x1 + xq
y = y1 + yq
z = z1 + zq

圆环面螺线管(矩阵原理):

#http://xuxzmail.blog.163.com/blog/static/25131916200976114621705/
#Toroidal spiral

vertices = D1: D2:

u = from  to (*PI) D1
v = from  to (*PI) D2

r =
n = 

xq = (r+sin(*v))*cos(v)
yq = (r+sin(*v))*sin(v)
zq = cos(n*v)

#由线变管

xd = array_difference(xq, )
yd = array_difference(yq, )
zd = array_difference(zq, )

len = sqrt(xd*xd + yd*yd + zd*zd)
len = max(len, 0.00001)
a = xd/len
b = yd/len
c = zd/len

len = sqrt(b*b + c*c)
len = max(len, 0.00001)
m = c/len
n = -b/len

i = b*n - c*m
j = - a*n
k = a*m

radius = 0.2

x0 = cos(u) * radius
z0 = sin(u) * radius

x = x0*i + xq
y = x0*j + z0*m + yq
z = x0*k + z0*n + zq

这是我所写的最复杂的脚本代码了,为了写这种脚本,我不惜将脚本解析器实现对自定义变量的支持.

不过这两种写法都会有点瑕疵,因为在某个情况下拐点时,会出现扭曲,这种瑕疵似乎很难解决.

再举个节的例子:

line_torus_knot(37)

vertices =
t = from  to (*PI)

p =
q = 

r =  + cos(q/p*t)

x = r*sin(t)
y = sin(q/p*t)
z = r*cos(t)

r = 0.5 + 0.5*sin(t)
g = 0.5 + 0.5*y
b = 0.5 + 0.5*cos(t)

pipe_torus_knot(37)极坐标原理

vertices = D1: D2:

u = from  to (*PI) D1
v = from  to (*PI) D2

p =
q = 

r =  + cos(q/p*v)

xq = r*sin(v)
yq = sin(q/p*v)
zq = r*cos(v)

r = 0.5 + 0.5*sin(v)
g = 0.5 + 0.5*yq
b = 0.5 + 0.5*cos(v)

#由线变管

xd = array_difference(xq, )
yd = array_difference(yq, )
zd = array_difference(zq, )

len = sqrt(xd*xd + yd*yd + zd*zd)
a = xd/len
b = yd/len
c = zd/len

radius = 0.2

ac = sqrt(a*a + c*c)
angleY = atan2(ac, b)
angleXZ = atan2(a,c)

m = cos(u) * radius
n = sin(u) * radius

e = sin(angleY)
f = cos(angleY)

g = sin(angleXZ)
h = cos(angleXZ)

x0 = m
y0 = -n*e
z0 = n*f

x1 = x0*h + z0*g
y1 = y0
z1 = -x0*g + z0*h

x = x1 + xq
y = y1 + yq
z = z1 + zq

pipe_torus_knot(37)矩阵原理

vertices = D1: D2:

u = from  to (*PI) D1
v = from  to (*PI) D2

p =
q = 

r =  + cos(q/p*v)

xq = r*sin(v)
yq = sin(q/p*v)
zq = r*cos(v)

r = 0.5 + 0.5*sin(v)
g = 0.5 + 0.5*yq
b = 0.5 + 0.5*cos(v)

#由线变管

xd = array_difference(xq, )
yd = array_difference(yq, )
zd = array_difference(zq, )

len = sqrt(xd*xd + yd*yd + zd*zd)
px = xd/len
py = yd/len
pz = zd/len

len = sqrt(py*py + pz*pz)
len = max(len, 0.00001)
m = pz/len
n = -py/len

i = py*n - pz*m
j = - px*n
k = px*m

radius = 0.2

x0 = cos(u) * radius
z0 = sin(u) * radius

x = x0*i + xq
y = x0*j + z0*m + yq
z = x0*k + z0*n + zq