Transformation article

animation/animation.py

@@ -30,32 +30,6 @@ class Animation(object):
     def __str__(self):
         return self.name
 
-    def get_points_and_rgbs(self):
-        """
-        It is the responsibility of this class to only emit points within
-        the space.  Returns np array of points and corresponding np array 
-        of rgbs
-        """
-        #TODO, I don't think this should be necessary.  This should happen 
-        #under the individual mobjects.  
-        points = self.mobject.points
-        rgbs   = self.mobject.rgbs
-        #Filters out what is out of bounds.
-        admissibles = (abs(points[:,0]) < self.restricted_width) * \
-                      (abs(points[:,1]) < self.restricted_height)
-        for filter_function in self.filter_functions:
-            admissibles *= ~filter_function(points)
-        if any(self.spatial_center):
-            points += self.spatial_center
-            #Filter out points pushed off the edge
-            admissibles *= (abs(points[:,0]) < SPACE_WIDTH) * \
-                           (abs(points[:,1]) < SPACE_HEIGHT)
-        if rgbs.shape[0] < points.shape[0]:
-            #TODO, this shouldn't be necessary, find what's happening.
-            points = points[:rgbs.shape[0], :]
-            admissibles = admissibles[:rgbs.shape[0]]
-        return points[admissibles, :], rgbs[admissibles, :]
-
     def update(self, alpha):
         if alpha < 0:
             alpha = 0.0

animation/meta_animations.py

@@ -4,6 +4,7 @@ import itertools as it
 from helpers import *
 from animation import Animation
 from transform import Transform
+from mobject import Mobject
 
 
 class DelayByOrder(Animation):
@@ -43,6 +44,7 @@ class TransformAnimations(Transform):
         "alpha_func" : squish_alpha_func(smooth)
     }
     def __init__(self, start_anim, end_anim, **kwargs):
+        digest_config(self, kwargs, locals())
         if "run_time" in kwargs:
             self.run_time = kwargs.pop("run_time")
         else:

animation/simple_animations.py

@@ -11,14 +11,26 @@ from transform import Transform
 class Rotating(Animation):
     DEFAULT_CONFIG = {
         "axes"       : [RIGHT, UP],
+        "axis"       : None,
         "radians"    : 2*np.pi,
         "run_time"   : 20.0,
         "alpha_func" : None,
+        "in_place"   : True,
     }
     def update_mobject(self, alpha):
-        self.mobject.points = self.starting_mobject.points
-        for axis in self.axes:
-            self.mobject.rotate(self.radians * alpha, axis)
+        axes = [self.axis] if self.axis is not None else self.axes
+        families = [
+            self.mobject.get_full_submobject_family(),
+            self.starting_mobject.get_full_submobject_family()
+        ]
+        for mob, start in zip(*families):
+            mob.points = np.array(start.points)
+        if self.in_place:
+            method = self.mobject.rotate_in_place
+        else:
+            method = self.mobject.rotate           
+        method(alpha*self.radians, axes = axes)      
+
 
 class FadeOut(Animation):
     def update_mobject(self, alpha):

animation/transform.py

@@ -103,9 +103,17 @@ class ApplyMethod(Transform):
         )
 
 class Rotate(ApplyMethod):
-    def __init__(self, mobject, angle = np.pi, **kwargs):
+    DEFAULT_CONFIG = {
+        "in_place" : False,
+    }
+    def __init__(self, mobject, angle = np.pi, axis = OUT, **kwargs):
         kwargs["interpolation_function"] = path_along_arc(angle)
-        ApplyMethod.__init__(self, mobject.rotate, angle, **kwargs)
+        digest_config(self, kwargs, locals())
+        if self.in_place:
+            method = mobject.rotate_in_place
+        else:
+            method = mobject.rotate
+        ApplyMethod.__init__(self, method, angle, axis, **kwargs)
 
 
 class ApplyPointwiseFunction(ApplyMethod):

mobject/mobject.py

@@ -120,12 +120,17 @@ class Mobject(object):
             mob.points *= scale_factor
         return self
 
-    def rotate(self, angle, axis = OUT):
-        t_rotation_matrix = np.transpose(rotation_matrix(angle, axis))
+    def rotate(self, angle, axis = OUT, axes = []):
+        if len(axes) == 0:
+            axes = [axis]
+        rot_matrix = np.identity(self.DIM)
+        for axis in axes:
+            rot_matrix = np.dot(rot_matrix, rotation_matrix(angle, axis))
+        t_rot_matrix = np.transpose(rot_matrix)
         for mob in self.get_full_submobject_family():
-            mob.points = np.dot(mob.points, t_rotation_matrix)
+            mob.points = np.dot(mob.points, t_rot_matrix)
             if mob.has_normals:
-                mob.unit_normals = np.dot(mob.unit_normals, t_rotation_matrix)
+                mob.unit_normals = np.dot(mob.unit_normals, t_rot_matrix)
         return self
 
     def stretch(self, factor, dim):
@@ -165,6 +170,8 @@ class Mobject(object):
 
     def filter_out(self, condition):
         for mob in self.get_full_submobject_family():
+            if len(mob.points) == 0:
+                continue
             to_eliminate = ~np.apply_along_axis(condition, 1, mob.points)
             mob.points = mob.points[to_eliminate]
             mob.rgbs = mob.rgbs[to_eliminate]
@@ -205,8 +212,8 @@ class Mobject(object):
         self.shift(center)
         return self
 
-    def rotate_in_place(self, angle, axis = OUT):
-        self.do_in_place(self.rotate, angle, axis)
+    def rotate_in_place(self, angle, axis = OUT, axes = []):
+        self.do_in_place(self.rotate, angle, axis, axes)
         return self
 
     def scale_in_place(self, scale_factor):
@@ -314,13 +321,10 @@ class Mobject(object):
             return 0
 
     def get_merged_array(self, array_attr):
-        return reduce(
-            lambda a1, a2 : np.append(a1, a2, axis = 0),
-            [getattr(self, array_attr)] + [
-                mob.get_merged_array(array_attr) 
-                for mob in self.sub_mobjects
-            ]
-        )
+        result = np.zeros((0, self.DIM))
+        for mob in self.get_full_submobject_family():
+            result = np.append(result, getattr(mob, array_attr), 0)
+        return result
 
     def get_all_points(self):
         return self.get_merged_array("points")

topics/number_line.py

@@ -229,6 +229,7 @@ class XYZAxes(Mobject1D):
         self.z_axis = self.x_axis.copy().rotate(np.pi/2, DOWN)
 
         self.digest_mobject_attrs()
+        self.pose_at_angle()
 
 
 class SpaceGrid(Mobject1D):
@@ -246,6 +247,7 @@ class SpaceGrid(Mobject1D):
                 start = np.array([a, b, -self.radius])[perm]
                 end   = np.array([a, b, self.radius])[perm]
                 self.add_line(start, end)
+        self.pose_at_angle()
 
 
 

transform_article.py

@@ -7,7 +7,9 @@ def half_plane():
         x_radius = SPACE_WIDTH/2,
         x_unit_to_spatial_width  = 0.5,
         y_unit_to_spatial_height = 0.5,
-        density = 2*DEFAULT_POINT_DENSITY_1D,
+        x_faded_line_frequency = 0,
+        y_faded_line_frequency = 0,
+        density = 4*DEFAULT_POINT_DENSITY_1D,
     )
     plane.add_coordinates(
         x_vals = range(-6, 7, 2),
@@ -71,12 +73,12 @@ class SingleVariableFunction(Scene):
 
 class LineToPlaneFunction(Scene):
     args_list = [
-        (lambda x : (np.cos(x), 0.5*x*np.sin(x)), "Swirl", {}),
-        (lambda x : (np.cos(x), 0.5*x*np.sin(x)), "Swirl", {
-            "0" : 0, 
-            "\\frac{\\pi}{2}" : np.pi/2, 
-            "\\pi" : np.pi
-        })        
+        (lambda x : (np.cos(x), 0.5*x*np.sin(x)), "Swirl", []),
+        (lambda x : (np.cos(x), 0.5*x*np.sin(x)), "Swirl", [
+            ("0", "(1, 0)", 0),
+            ("\\frac{\\pi}{2}",  "(0, \\pi / 4)", np.pi/2),
+            ("\\pi", "(-1, 0)", np.pi),
+        ])        
     ]
 
     @staticmethod
@@ -97,6 +99,10 @@ class LineToPlaneFunction(Scene):
         line.add_numbers(*range(0, 14, 2))
         divider = Line(SPACE_HEIGHT*UP, SPACE_HEIGHT*DOWN)
         plane = half_plane()
+        plane.sub_mobjects = []
+        plane.filter_out(
+            lambda (x, y, z) : abs(x) > 0.1 and abs(y) > 0.1
+        )
         plane.shift(0.5*SPACE_WIDTH*RIGHT)
         self.add(line, divider, plane)
 
@@ -110,7 +116,7 @@ class LineToPlaneFunction(Scene):
         anims = [Transform(line_copy, target, **anim_config)]
 
         colors = iter([BLUE_B, GREEN_D, RED_D])
-        for tex, number in numbers_to_follow.items():
+        for input_tex, output_tex, number in numbers_to_follow:
             center = line.number_to_point(number)
             dot = Dot(center, color = colors.next())
             anims.append(ApplyMethod(
@@ -118,7 +124,7 @@ class LineToPlaneFunction(Scene):
                 point_function(center) - center, 
                 **anim_config 
             ))
-            label = TexMobject(tex)
+            label = TexMobject(input_tex)
             label.shift(center + 2*UP)
             arrow = Arrow(label, dot)
             self.add(label)
@@ -131,6 +137,17 @@ class LineToPlaneFunction(Scene):
         self.play(*anims)
         self.dither()
 
+        for input_tex, output_tex, number in numbers_to_follow:
+            point = plane.num_pair_to_point(func(number))
+            label = TexMobject(output_tex)
+            side_shift = LEFT if number == np.pi else RIGHT
+            label.shift(point, 2*UP, side_shift)
+            arrow = Arrow(label, point)
+            self.add(label)
+            self.play(ShowCreation(arrow))
+            self.dither(2)
+            self.remove(arrow, label)
+
 class PlaneToPlaneFunctionSeparatePlanes(Scene):
     args_list = [
         (lambda (x, y) : (x**2+y**2, x**2-y**2), "Quadratic")
@@ -185,9 +202,119 @@ class PlaneToPlaneFunction(Scene):
         self.play(ApplyPointwiseFunction(point_function, plane, **anim_config))
         self.dither(3)
 
-
-
-
+class PlaneToLineFunction(Scene):
+    args_list = [
+        (lambda (x, y) : x**2 + y**2, "Bowl"),
+    ]
+
+    @staticmethod
+    def args_to_string(func, name):
+        return name
+
+    def construct(self, func, name):
+        line = NumberLine(
+            color = GREEN,
+            unit_length_to_spatial_width = 0.5,
+            tick_frequency = 1,
+            number_at_center = 6,
+            numerical_radius = 6,
+            numbers_with_elongated_ticks = [0, 12],
+        ).to_edge(RIGHT)
+        line.add_numbers()
+        plane = half_plane().to_edge(LEFT, buff = 0)
+
+        divider = Line(SPACE_HEIGHT*UP, SPACE_HEIGHT*DOWN)
+        line_left = line.number_to_point(0)
+        def point_function(point):
+            shifter = 0.5*SPACE_WIDTH*RIGHT
+            return func((point+shifter)[:2])*RIGHT + line_left
+
+        self.add(line, plane, divider)
+        self.dither()
+        plane.sub_mobjects = []
+        self.play(ApplyPointwiseFunction(point_function, plane))
+        self.dither()
+
+
+
+class PlaneToSpaceFunction(Scene):
+    args_list = [
+        (lambda (x, y) : (x*x, x*y, y*y), "Quadratic"),
+    ]
+
+    @staticmethod
+    def args_to_string(func, name):
+        return name
+
+    def construct(self, func, name):
+        plane = half_plane().shift(0.5*SPACE_WIDTH*LEFT)
+        divider = Line(SPACE_HEIGHT*UP, SPACE_HEIGHT*DOWN)
+        axes = XYZAxes()
+        axes.filter_out(lambda p : np.linalg.norm(p) > 3)
+        rot_kwargs = {
+            "run_time" : 3,
+            "radians"  : 0.3*np.pi,
+            "axis"     : [0.1, 1, 0.1],
+        }
+        axes.to_edge(RIGHT).shift(DOWN)        
+        dampening_factor = 0.1
+        def point_function((x, y, z)):
+            return dampening_factor*np.array(func((x, y)))
+        target = NumberPlane().apply_function(point_function)
+        target.highlight("yellow")
+        target.shift(axes.get_center())
+
+        self.add(plane, divider, axes)
+        self.play(Rotating(axes, **rot_kwargs))
+
+        target.rotate_in_place(rot_kwargs["radians"])
+        self.play(
+            TransformAnimations(
+                Animation(plane.copy()),
+                Rotating(target, **rot_kwargs),
+                alpha_func = smooth
+            ),
+            Rotating(axes, **rot_kwargs)
+        )
+        axes.add(target)
+        self.clear()
+        self.add(plane, divider, axes)
+        self.play(Rotating(axes, **rot_kwargs))
+        self.clear()
+        for i in range(5):
+            self.play(Rotating(axes, **rot_kwargs))
+
+
+class SpaceToSpaceFunction(Scene):
+    args_list = [
+        (lambda (x, y, z) : (y*z, x*z, x*y), "Quadratic"),
+    ]
+
+    @staticmethod
+    def args_to_string(func, name):
+        return name
+
+    def construct(self, func, name):
+        space = SpaceGrid()
+        rot_kwargs = {
+            "run_time" : 10,
+            "radians"  : 2*np.pi/5,
+            "axis"     : [0.1, 1, 0.1],
+            "in_place" : False,
+        }
+        axes = XYZAxes()
+        target = space.copy().apply_function(func)
+
+        self.play(
+            TransformAnimations(
+                Rotating(space, **rot_kwargs),
+                Rotating(target, **rot_kwargs),
+                alpha_func = squish_alpha_func(smooth, 0.3, 0.7)
+            ),
+            Rotating(axes, **rot_kwargs)
+        )
+        axes.add(space)
+        self.play(Rotating(axes, **rot_kwargs))