NodeBox

Mark Meyer | Parametric surfaces | trefoil knot

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 = 150
 
# --- TREFOIL 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).
r = 120
def x(u, v):
    return  r * sin(3*u) / (2 + cos(v))
def y(u, v):
    return r * (sin(u) + 2 * sin(2*u)) / (2 + cos(v + 2*pi/3))
def z(u, v):
    return r/2 * (cos(u) - 2 * cos(2*u)) \
               * (2 + cos(v)) \
               * (2 + cos(v + 2*pi/3)) / 4
 
def fit_to_domain(u, v):
    u = -pi + u * 2*pi / n
    v = -pi + v * 2*pi / (n-1)
    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, 0.00, -pi, pi)
var("rot_y", NUMBER, pi/2, -pi, pi)
var("rot_z", NUMBER, 0.00, -pi, pi)
 
# --- LIGHTING SLIDERS -----------------------------------------------------
 
var("light_x", NUMBER,  0.0, -1.0, 1.0)
var("light_y", NUMBER, -1.0, -1.0, 1.0)
var("light_z", NUMBER,  0.5, -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