### Mark Meyer | Parametric surfaces | torus

```size(550, 550)
background(0)
nofill()
stroke(0.6, 0.6, 0.7, 0.35)
strokewidth(0.5)

translate(WIDTH/2, HEIGHT/2)

# n defines how many vertices we find: the number will be n^2.
# Large values of n can take a while to render!
n = 50

# --- TORUS EQUATION --------------------------------------------------------

# Define parametric equations for each axis.
# Although we define an z coordinate,
# it is only used in transformations and shading;
# it is not used in the drawing
# (although it could be if you wanted to add perspective).
r1 = 140
r2 = 50
def x(u, v): return (r1 + r2 * cos(v)) * cos(u)
def y(u, v): return (r1 + r2 * cos(v)) * sin(u)
def z(u, v): return r2 * sin(v)

def fit_to_domain(u, v):
u = 1.0 * u/n * pi*2
v = 1.0 * v/n * pi*2
return u, v

from Numeric import *
def matrix(u, v, index):
u, v = fit_to_domain(u, v)
return where(index==0, x(u, v),
where(index==1, y(u,v),
where(index==2,  z(u,v), 1)
)
)

# --- MATRIX ROTATION ------------------------------------------------------

def rotate_matrix(matrix, x, y, z):
rot_x = array(
([1, 0, 0],
[0, cos(x), -sin(x)],
[0, sin(x), cos(x)]
), Float
)
rot_y = array(
([cos(y), 0, sin(y)],
[0, 1, 0],
[-sin(y), 0, cos(y)]
), Float
)
rot_z = array(
([cos(z),  -sin(z), 0],
[sin(z), cos(z), 0],
[0, 0, 1]
), Float
)
matrix = matrixmultiply(matrix, rot_x)
matrix = matrixmultiply(matrix, rot_y)
matrix = matrixmultiply(matrix, rot_z)
return matrix

# --- VECTOR NORMALIZATION -------------------------------------------------

def length(vector):
return sqrt(vector[0]**2 + vector[1]**2 + vector[2]**2)

def normalize_vector(face):
""" Calculate the normal vector between -1 and 1 for a flat surface """
norm = cross_product((face[0,0] - face[1,1]), (face[1,0] - face[0,1]))
return norm / length(norm)

# --- ORTHOGRAPHIC PROJECTION ----------------------------------------------

def project(rows):
"""
Go through the array and draw rectangles.
There is probably a more efficent way to do this.
"""
for i in range(len(rows) -1):
for j in range(len(rows[i])-1):
face = rows[i:i+2, j:j+2]
normal = normalize_vector(face)
light_angle = (dot(normal, light))
if (light_angle < 0):
light_angle = 0
fill(
light_angle + 0.02,
light_angle + 0.04,
light_angle + 0.08,
1.0
)
# Draw only forward facing facets.
if (dot(view, normal) > 0):
beginpath(
face[0, 0, 0],
face[0, 0, 1]
)
lineto(
face[1, 0, 0],
face[1, 0, 1]
)
lineto(
face[1, 1, 0],
face[1, 1, 1]
)
lineto(
face[0, 1, 0],
face[0, 1, 1]
)
path = endpath()
# Save z with path for later z sorting.
path.z = face[1,0,2]

# --- ROTATION SLIDERS -----------------------------------------------------

var("rot_x", NUMBER, 1.00, -pi, pi)
var("rot_y", NUMBER, pi/4, -pi, pi)
var("rot_z", NUMBER, -e,   -pi, pi)

# --- LIGHTING SLIDERS -----------------------------------------------------

var("light_x", NUMBER,  0.50, -1.0, 1.0)
var("light_y", NUMBER, -0.60, -1.0, 1.0)
var("light_z", NUMBER, -0.25, -1.0, 1.0)

light = array([light_x, light_y, light_z], Float)

# --- FACET CULLING --------------------------------------------------------
# Draw only forward facing facets.
view = array([0, 0, 1], Float)

sphere = fromfunction(matrix, (n+1, n+1, 3))
sphere = rotate_matrix(sphere, rot_x, rot_y, rot_z)
project(sphere)

# z-sort to avoid problems with overlapping surfaces:
sorted_grobs = list(canvas)
sorted_grobs.sort(lambda p1, p2: cmp(p1.z, p2.z))
canvas._grobs = sorted_grobs```