Given any 3D graphics object one can compute a raytraced representation by typing show(viewer='tachyon'). For example, we draw two translucent spheres that contain a red tube, and render the result using Tachyon.
sage: S = sphere(opacity=0.8, aspect_ratio=[1,1,1])
sage: L = line3d([(0,0,0),(2,0,0)], thickness=10, color='red')
sage: M = S + S.translate((2,0,0)) + L
sage: M.show(viewer='tachyon')
One can also directly control Tachyon, which gives a huge amount of flexibility. For example, here we directly use Tachyon to draw 3 spheres on the coordinate axes. Notice that the result is gorgeous:
sage: t = Tachyon(xres=500,yres=500, camera_center=(2,0,0))
sage: t.light((4,3,2), 0.2, (1,1,1))
sage: t.texture('t2', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1,0,0))
sage: t.texture('t3', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(0,1,0))
sage: t.texture('t4', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(0,0,1))
sage: t.sphere((0,0.5,0), 0.2, 't2')
sage: t.sphere((0.5,0,0), 0.2, 't3')
sage: t.sphere((0,0,0.5), 0.2, 't4')
sage: t.show()
AUTHOR:
TODO:
Create a scene the can be rendered using the Tachyon ray tracer.
INPUT:
OUTPUT: A Tachyon 3d scene.
Note that the coordinates are by default such that is
up, positive
is to the {left} and
is toward
you. This is not oriented according to the right hand rule.
EXAMPLES: Spheres along the twisted cubic.
sage: t = Tachyon(xres=512,yres=512, camera_center=(3,0.3,0))
sage: t.light((4,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2,diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: k=0
sage: for i in srange(-1,1,0.05):
... k += 1
... t.sphere((i,i^2-0.5,i^3), 0.1, 't%s'%(k%3))
...
sage: t.show()
Another twisted cubic, but with a white background, got by putting infinite planes around the scene.
sage: t = Tachyon(xres=512,yres=512, camera_center=(3,0.3,0), raydepth=8)
sage: t.light((4,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2,diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.plane((0,0,-1), (0,0,1), 'white')
sage: t.plane((0,-20,0), (0,1,0), 'white')
sage: t.plane((-20,0,0), (1,0,0), 'white')
sage: k=0
sage: for i in srange(-1,1,0.05):
... k += 1
... t.sphere((i,i^2 - 0.5,i^3), 0.1, 't%s'%(k%3))
... t.cylinder((0,0,0), (0,0,1), 0.05,'t1')
...
sage: t.show()
Many random spheres:
sage: t = Tachyon(xres=512,yres=512, camera_center=(2,0.5,0.5), look_at=(0.5,0.5,0.5), raydepth=4)
sage: t.light((4,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: k=0
sage: for i in range(100):
... k += 1
... t.sphere((random(),random(), random()), random()/10, 't%s'%(k%3))
...
sage: t.show()
Points on an elliptic curve, their height indicated by their height above the axis:
sage: t = Tachyon(camera_center=(5,2,2), look_at=(0,1,0))
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(0,1,0))
sage: t.texture('t2', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(0,0,1))
sage: E = EllipticCurve('37a')
sage: P = E([0,0])
sage: Q = P
sage: n = 100
sage: for i in range(n): # increase 20 for a better plot
... Q = Q + P
... t.sphere((Q[1], Q[0], ZZ(i)/n), 0.1, 't%s'%(i%3))
...
sage: t.show()
A beautiful picture of rational points on a rank 1 elliptic curve.
sage: t = Tachyon(xres=1000, yres=800, camera_center=(2,7,4), look_at=(2,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.light((10,-3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: E = EllipticCurve('37a')
sage: P = E([0,0])
sage: Q = P
sage: n = 100
sage: for i in range(n):
... Q = Q + P
... c = i/n + .1
... t.texture('r%s'%i,color=(float(i/n),0,0))
... t.sphere((Q[0], -Q[1], .01), .04, 'r%s'%i)
...
...
sage: t.show() # long time, e.g., 10-20 seconds
A beautiful spiral.
sage: t = Tachyon(xres=800,yres=800, camera_center=(2,5,2), look_at=(2.5,0,0))
sage: t.light((0,0,100), 1, (1,1,1))
sage: t.texture('r', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1,0,0))
sage: for i in srange(0,50,0.1):
... t.sphere((i/10,sin(i),cos(i)), 0.05, 'r')
...
sage: t.texture('white', color=(1,1,1), opacity=1, specular=1, diffuse=1)
sage: t.plane((0,0,-100), (0,0,-100), 'white')
sage: t.show()
Plots a space curve as a series of spheres and finite cylinders. Example (twisted cubic)
sage: f = lambda t: (t,t^2,t^3)
sage: t = Tachyon(camera_center=(5,0,4))
sage: t.texture('t')
sage: t.light((-20,-20,40), 0.2, (1,1,1))
sage: t.parametric_plot(f,-5,5,'t',min_depth=6)
INPUT:
Plots a function by constructing a mesh with nonstandard sampling
density without gaps. At very high resolutions (depths 10) it
becomes very slow. Cython may help. Complexity is approx.
. This algorithm has been optimized for
speed, not memory - values from f(x,y) are recycled rather than
calling the function multiple times. At high recursion depth, this
may cause problems for some machines.
Flat Triangles:
sage: t = Tachyon(xres=512,yres=512, camera_center=(4,-4,3),viewdir=(-4,4,-3), raydepth=4)
sage: t.light((4.4,-4.4,4.4), 0.2, (1,1,1))
sage: def f(x,y): return float(sin(x*y))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.1, opacity=1.0, color=(1.0,0,0))
sage: t.plot(f,(-4,4),(-4,4),"t0",max_depth=5,initial_depth=3, num_colors=60) # increase min_depth for better picture
sage: t.show()
Plotting with Smooth Triangles (requires explicit gradient function):
sage: t = Tachyon(xres=512,yres=512, camera_center=(4,-4,3),viewdir=(-4,4,-3), raydepth=4)
sage: t.light((4.4,-4.4,4.4), 0.2, (1,1,1))
sage: def f(x,y): return float(sin(x*y))
sage: def g(x,y): return ( float(y*cos(x*y)), float(x*cos(x*y)), 1 )
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.1, opacity=1.0, color=(1.0,0,0))
sage: t.plot(f,(-4,4),(-4,4),"t0",max_depth=5,initial_depth=3, grad_f = g) # increase min_depth for better picture
sage: t.show()
Preconditions: f is a scalar function of two variables, grad_f is None or a triple-valued function of two variables, min_x != max_x, min_y != max_y
sage: f = lambda x,y: x*y
sage: t = Tachyon()
sage: t.plot(f,(2.,2.),(-2.,2.),'')
...
ValueError: Plot rectangle is really a line. Make sure min_x != max_x and min_y != max_y.
INPUT:
INPUT:
EXAMPLES: We draw an infinite checkboard:
sage: t = Tachyon(camera_center=(2,7,4), look_at=(2,0,0))
sage: t.texture('black', color=(0,0,0), texfunc=1)
sage: t.plane((0,0,0),(0,0,1),'black')
sage: t.show()
INPUT:
EXAMPLES: We draw a scene with 4 sphere that illustrates various uses of the texture command:
sage: t = Tachyon(camera_center=(2,5,4), look_at=(2,0,0), raydepth=6)
sage: t.light((10,3,4), 1, (1,1,1))
sage: t.texture('mirror', ambient=0.05, diffuse=0.05, specular=.9, opacity=0.9, color=(.8,.8,.8))
sage: t.texture('grey', color=(.8,.8,.8), texfunc=3)
sage: t.plane((0,0,0),(0,0,1),'grey')
sage: t.sphere((4,-1,1), 1, 'mirror')
sage: t.sphere((0,-1,1), 1, 'mirror')
sage: t.sphere((2,-1,1), 0.5, 'mirror')
sage: t.sphere((2,1,1), 0.5, 'mirror')
sage: show(t)
Converts vector information to a space-separated string.
EXAMPLES:
sage: from sage.plot.plot3d.tachyon import tostr
sage: tostr((1,1,1))
' 1.0 1.0 1.0 '
sage: tostr('2 3 2')
'2 3 2'