From 913b78d10c5ca9ace968d7bf9144b540a7743ff7 Mon Sep 17 00:00:00 2001 From: sujaygarlanka Date: Sat, 10 May 2025 20:13:28 -0400 Subject: [PATCH 1/2] Add KD tree animation --- MotionPlanningHigherDimensions.ipynb | 3 +- animations/code/motion_planning/kd_tree.py | 542 ++++++++++++++++++++ animations/media/motion_planning/KDTree.mp4 | Bin 0 -> 1683603 bytes 3 files changed, 544 insertions(+), 1 deletion(-) create mode 100644 animations/code/motion_planning/kd_tree.py create mode 100644 animations/media/motion_planning/KDTree.mp4 diff --git a/MotionPlanningHigherDimensions.ipynb b/MotionPlanningHigherDimensions.ipynb index 6af3c94..c8b0dfe 100644 --- a/MotionPlanningHigherDimensions.ipynb +++ b/MotionPlanningHigherDimensions.ipynb @@ -961,7 +961,7 @@ "source": [ "#### k-d trees\n", "\n", - "A *$k$-d tree* data structure is a spatial hierarchy that recursively divides a Cartesian space $\\mathbb{R}^k$ into regions by splitting each region by a hyperplane. *The $k$ here refers to the number of dimensions in the space, not the $k$ in the kNN query. In the following, let us revert back to our original notation where dimensionality is denoted $d$.* Each hyperplane is aligned to one of the $d$ primary axes. An illustration of a k-d tree is shown below.\n", + "A *$k$-d tree* data structure is a spatial hierarchy that recursively divides a Cartesian space $\\mathbb{R}^k$ into regions by splitting each region by a hyperplane. *The $k$ here refers to the number of dimensions in the space, not the $k$ in the kNN query. In the following, let us revert back to our original notation where dimensionality is denoted $d$.* Each hyperplane is aligned to one of the $d$ primary axes. An illustration and explanatory video of a k-d tree is shown below.\n", "\n", "***************\n", "\n", @@ -975,6 +975,7 @@ "(a) $k$-d trees recursively divide a space into rectilinear regions. Each leaf of the tree contains a set of points contained in that region. (b) For a nearest-neighbor query (blue point), the leaf containing the point is reached first, and the closest point in the leaf is found (blue circle). This forms an upper bound on the nearest neighbor's distance, and any leaves further than this distance will be pruned. (c) Only one more leaf is visited before the nearest neighbor is found.\n", "\n", "\n", + "\n", "***************\n", "\n", "More formally, the binary tree $T_{kd}$ is composed of nodes $N$. Each leaf node contains a list of contained points `pts`, and each non-leaf node contains a split dimension `dim` and split value `value`. Non-leaf nodes have exactly two children $C^-$, and $C^+$, and all points $q \\in \\mathbb{R}^d$ such that $q_{dim} < value$ belong to the _negative_ child $C^-$, while points such that $q_{dim} \\geq value$ belong to the _positive_ child $C^+$.\n", diff --git a/animations/code/motion_planning/kd_tree.py b/animations/code/motion_planning/kd_tree.py new file mode 100644 index 0000000..9cebc24 --- /dev/null +++ b/animations/code/motion_planning/kd_tree.py @@ -0,0 +1,542 @@ +from manimlib import * +import numpy as np + +manim_config.camera.background_color = WHITE + +class KDTreeNode: + def __init__(self, name, dim=None, point=None, leaf=False): + self.name = name + if not leaf: + self.dim = dim + self.val = [point.local_x, point.local_y][dim] + self.dot = point.dot + self.leaf = leaf + self.children = [] + self.graph_node = None + self.edge = None + self.areas = [] + + def __repr__(self): + return f"KDTreeNode({self.point})" + +class Point(): + def __init__(self, x, y, axes): + self.axes = axes + self.local_x = x + self.local_y = y + self.dot = Dot(point=axes.c2p(x, y), fill_color=YELLOW, stroke_color=BLACK, stroke_width=3) + self.dot.set_z_index(10) + self.world_x = self.dot.get_center()[0] + self.world_y = self.dot.get_center()[1] + +class KDTree(Scene): + def construct(self): + self.text_obj = None + self._title_screen("Create KD-Tree") + self.points, self.axes = self._create_scene() + kd_tree = self._create_kd_tree() + self._title_screen("Find Nearest Neighbor using the KD-Tree") + point = Point(5.9, 4.3, self.axes) + self._find_nearest_neighbor(kd_tree, point) + + def _find_nearest_neighbor(self, kd_tree, point): + # Create a dot for the point + point.dot.set_color("#0000FF") + # Create a dashed circle + circle = Circle(radius=0.01, stroke_color="#0000FF", stroke_width=3) + circle.move_to(point.dot.get_center()) + line = Line( + start=point.dot.get_center(), + end=point.dot.get_center(), + color="#0000FF", + stroke_width=3, + ) + + self.play(FadeIn(circle), FadeIn(line), FadeIn(point.dot), run_time=1) + + # KD search algorithm + def dfs(node, nn): + if node.leaf: + self.play( + node.graph_node.animate.set_stroke(GREEN), + ) + if node.name == "E-Left": + self._write_info(""" + Now, we are at a leaf node, we iterate through the points in the node store the closest one (highlighted red). + """, {"red": RED}) + self._delete_info() + for p in node.children: + # Calculate distance + distance = ((point.local_x - p.local_x)**2 + (point.local_y - p.local_y)**2)**0.5 + if min(distance, nn[0]) == distance: + if nn[1] is not None: + # Reset the color of the previous nearest neighbor + self.play( + nn[1].dot.animate.set_fill(YELLOW, opacity=1), + ) + nn[0] = distance + nn[1] = p + # Highlight the nearest neighbor and scale the circle + # Distance between point and p + actual_distance = ((p.dot.get_center()[0] - point.dot.get_center()[0])**2 + (p.dot.get_center()[1] - point.dot.get_center()[1])**2)**0.5 + target_circle = Circle(radius=actual_distance, stroke_color="#0000FF", stroke_width=3).move_to(circle) + + self.play( + Transform(circle, target_circle), + line.animate.put_start_and_end_on( + point.dot.get_center(), + p.dot.get_center(), + ), + nn[1].dot.animate.set_fill(RED), + ) + self.play( + node.graph_node.animate.set_stroke(BLACK), + ) + return nn + + coord = [point.local_x, point.local_y] + first_child_index = 0 if node.val > coord[node.dim] else 1 + second_child_index = 1 - first_child_index + self.play( + node.graph_node.animate.set_stroke(GREEN), + node.edge.animate.set_stroke(GREEN), + ) + # Choose the side of the tree to search + if node.name == "A": + self._write_info(""" + To search the KD-Tree, we start at the root node and check if the (blue) point is on the left or right side of the tree. + If the point's value is less than the root's value in the x-direction, we go left. Otherwise, we go right. In this case, we go right. + """, {"x-direction": BLUE}) + self._delete_info() + elif node.name == "C": + self._write_info(""" + Now, if the point's value is less than the root's value in the y-direction, we go left. Otherwise, we go right. In this case, we go right. + """, {"y-direction": RED}) + self._delete_info() + self.play( + node.areas[first_child_index].animate.set_fill(opacity=0.5), + ) + self.play( + node.areas[first_child_index].animate.set_fill(opacity=0), + ) + nn = dfs(node.children[first_child_index], nn) + if node.name == "C": + self._write_info(""" + C is not closer than the nearest neighbor found so far, so we don't need to check the other side of the tree at C. We continue back up the tree. + """, {}) + self._delete_info() + if nn[0] > abs(coord[node.dim] - node.val): + if node.name == "E": + self._write_info(""" + As we go back up the tree, we check if the distance to the current node's value is less than the distance to the nearest neighbor found so far. + If it is, we check the other side of the tree like for this E node. This is because the nearest neighbor could be on the other side. + """, {}) + self._delete_info() + self.play( + node.areas[second_child_index].animate.set_fill(opacity=0.5), + ) + self.play( + node.areas[second_child_index].animate.set_fill(opacity=0), + ) + nn = dfs(node.children[second_child_index], nn) + self.play( + node.graph_node.animate.set_stroke(BLACK), + node.edge.animate.set_stroke(BLACK), + ) + return nn + dfs(kd_tree, [float("inf"), None]) + self._write_info(""" + The point highlighted in red is the nearest neighbor. + """, {"red": RED}) + + def _create_scene(self): + points = [ + (0.8, 2.8), + (1.9, 5.9), + (2.6, 3.4), + (2.8, 5.2), + (3.6, 6.4), + (3.9, 3.2), + (4.3, 0.3), + (4.7, 3.8), + (5.1, 1.9), + (5.8, 3.4), + (6.2, 3.0), + (6.2, 3.8), + (6.8, 1.6), + (7.1, 5.3) + ] + points.sort() + axes = Axes( + x_range=[0, 8, 1], + y_range=[0, 7, 1], + axis_config={"color": BLUE}, + height=5, + width=5, + ) + axes.set_color(BLACK) + axes.shift(LEFT*3.5) + self.play(ShowCreation(axes), run_time=3.0) + + points_dots = [Point(p[0], p[1], axes) for p in points] + self.play(*[FadeIn(point.dot) for point in points_dots], run_time=2.0) + return points_dots, axes + + def _create_area_highlight(self, p1, p2, p3, p4): + # Create a polygon to highlight the area + area = Polygon( + self.axes.c2p(p1[0], p1[1], 0.0), + self.axes.c2p(p2[0], p2[1], 0.0), + self.axes.c2p(p3[0], p3[1], 0.0), + self.axes.c2p(p4[0], p4[1], 0.0), + fill_color=RED, + fill_opacity=0.0, + stroke_width=0 + ) + self.add(area) + return area + + def _create_kd_tree(self): + # A + self._write_info( + """ + The first node of the KD-Tree has a value that is the median of the x values of the points. This divides the points into halves. + The points with x values less than the median (left side) and greater than the median (right side) will be associated with + the left and right child of the node, respectively. The point at the median is associated with the right child. + """, {"x": BLUE}) + a_p = self.points[5] + a_edge = self._draw_line((a_p.local_x, self.axes.y_range[0]), (a_p.local_x, self.axes.y_range[1]), "A", LEFT*0.2) + a_node = KDTreeNode("A", 0, a_p) + a_graph_node = self._draw_graph_node((3, 3), "A", 0, a_p.local_x) + a_node.graph_node = a_graph_node + a_node.edge = a_edge + # Left area + a_node.areas.append(self._create_area_highlight( + (self.axes.x_range[0], self.axes.y_range[0]), + (self.axes.x_range[0], self.axes.y_range[1]), + (a_p.local_x, self.axes.y_range[1]), + (a_p.local_x, self.axes.y_range[0]) + )) + # Right area + a_node.areas.append(self._create_area_highlight( + (a_p.local_x, self.axes.y_range[0]), + (a_p.local_x, self.axes.y_range[1]), + (self.axes.x_range[1], self.axes.y_range[1]), + (self.axes.x_range[1], self.axes.y_range[0]) + )) + self._delete_info() + + self._write_info( + """ + In the next step, the median of the points in the next dimension, the y-dimension, for each side of A is used for the B and C nodes. + """, {"y-dimension": RED}) + # B + b_p = self.points[3] + b_edge = self._draw_line((self.axes.x_range[0], b_p.local_y), (a_p.local_x, b_p.local_y), "B", DOWN*0.2) + b_node = KDTreeNode("B", 1, b_p) + b_node.edge = b_edge + b_graph_node = self._draw_graph_node((1.7, 1.5), "B", 1, b_p.local_y) + self._draw_arrow(a_graph_node, b_graph_node) + b_node.graph_node = b_graph_node + a_node.children.append(b_node) + # Bottom area + b_node.areas.append(self._create_area_highlight( + (self.axes.x_range[0], self.axes.y_range[0]), + (self.axes.x_range[0], b_p.local_y), + (a_p.local_x, b_p.local_y), + (a_p.local_x, self.axes.y_range[0]) + )) + # Top area + b_node.areas.append(self._create_area_highlight( + (self.axes.x_range[0], b_p.local_y), + (self.axes.x_range[0], self.axes.y_range[1]), + (a_p.local_x, self.axes.y_range[1]), + (a_p.local_x, b_p.local_y) + )) + + # C + c_p = self.points[9] + c_edge = self._draw_line((a_p.local_x, c_p.local_y), (self.axes.x_range[1], c_p.local_y), "C", RIGHT + DOWN*0.2) + c_node = KDTreeNode("C", 1, c_p) + c_node.edge = c_edge + c_graph_node = self._draw_graph_node((4.3, 1.5), "C", 1, c_p.local_y) + self._draw_arrow(a_graph_node, c_graph_node) + c_node.graph_node = c_graph_node + a_node.children.append(c_node) + # Bottom area + c_node.areas.append(self._create_area_highlight( + (a_p.local_x, self.axes.y_range[0]), + (a_p.local_x, c_p.local_y), + (self.axes.x_range[1], c_p.local_y), + (self.axes.x_range[1], self.axes.y_range[0]) + )) + # Top area + c_node.areas.append(self._create_area_highlight( + (a_p.local_x, c_p.local_y), + (a_p.local_x, self.axes.y_range[1]), + (self.axes.x_range[1], self.axes.y_range[1]), + (self.axes.x_range[1], c_p.local_y) + )) + self._delete_info() + + self._write_info( + """ + Now, for B, the next nodes are leaf nodes that CONTAIN the points because there are only 3 or less points on each side. + The same is not true for C, so it will continue adding nodes to split the points. + """, {}) + # B leaf node left + b_node_leaf_left = KDTreeNode("B-Left", leaf=True) + square = Square(side_length=0.6, stroke_color=BLACK, fill_opacity=0.0, stroke_width=3) + square.move_to(np.array([0.9, 0.0, 0.0])) + self.play(ShowCreation(square), run_time=1) + leaf_points = [ + self.points[0], + self.points[2] + ] + b_node_leaf_left.children = leaf_points + point_group = VGroup(*[point.dot.copy() for point in leaf_points]) + self.play(point_group.animate.move_to(square.get_center()).scale(0.3), run_time=1) + self._draw_arrow(b_graph_node, square) + b_node_leaf_left.graph_node = VGroup(square, point_group) + b_node.children.append(b_node_leaf_left) + + # B leaf node right + b_node_leaf_right = KDTreeNode("B-Right", leaf=True) + square = Square(side_length=0.6, stroke_color=BLACK, fill_opacity=0.0, stroke_width=3) + square.move_to(np.array([2.5, 0.0, 0.0])) + self.play(ShowCreation(square), run_time=1) + leaf_points = [ + self.points[1], + self.points[3], + self.points[4] + ] + b_node_leaf_right.children = leaf_points + point_group = VGroup(*[point.dot.copy() for point in leaf_points]) + self.play(point_group.animate.move_to(square.get_center()).scale(0.3), run_time=1) + self._draw_arrow(b_graph_node, square) + b_node_leaf_right.graph_node = VGroup(square, point_group) + b_node.children.append(b_node_leaf_right) + self._delete_info() + + self._write_info( + """ + To split C, we loop back to splitting amongst the x-dimension since we have already split the points in all the dimensions. + """, {"x-dimension": BLUE}) + # D + d_p = self.points[8] + d_edge = self._draw_line((d_p.local_x, c_p.local_y), (d_p.local_x, self.axes.y_range[0]), "D", LEFT*0.2) + d_node = KDTreeNode("D", 0, d_p) + d_node.edge = d_edge + d_graph_node = self._draw_graph_node((3.5, 0.0), "D", 0, d_p.local_x) + self._draw_arrow(c_graph_node, d_graph_node) + d_node.graph_node = d_graph_node + c_node.children.append(d_node) + # Left area + d_node.areas.append(self._create_area_highlight( + (a_p.local_x, self.axes.y_range[0]), + (a_p.local_x, c_p.local_y), + (d_p.local_x, c_p.local_y), + (d_p.local_x, self.axes.y_range[0]) + )) + # Right area + d_node.areas.append(self._create_area_highlight( + (d_p.local_x, self.axes.y_range[0]), + (d_p.local_x, c_p.local_y), + (self.axes.x_range[0], c_p.local_y), + (self.axes.x_range[0], self.axes.y_range[0]) + )) + + # E + e_p = self.points[11] + e_edge = self._draw_line((e_p.local_x, c_p.local_y), (e_p.local_x, self.axes.y_range[1]), "E", LEFT*0.2) + e_node = KDTreeNode("E", 0, e_p) + e_node.edge = e_edge + e_graph_node = self._draw_graph_node((5.1, 0.0), "E", 0, e_p.local_x) + self._draw_arrow(c_graph_node, e_graph_node) + e_node.graph_node = e_graph_node + c_node.children.append(e_node) + # Left area + e_node.areas.append(self._create_area_highlight( + (a_p.local_x, c_p.local_y), + (a_p.local_x, self.axes.y_range[1]), + (e_p.local_x, self.axes.y_range[1]), + (e_p.local_x, c_p.local_y) + )) + # Right area + e_node.areas.append(self._create_area_highlight( + (e_p.local_x, c_p.local_y), + (e_p.local_x, self.axes.y_range[1]), + (self.axes.x_range[1], self.axes.y_range[1]), + (self.axes.x_range[1], c_p.local_y) + )) + self._delete_info() + + self._write_info( + """ + Now, for D and E, the next nodes are leaf nodes that CONTAIN the points on each side. + """, {}) + # D leaf node left + d_node_leaf_left = KDTreeNode("D-Left", leaf=True) + square = Square(side_length=0.6, stroke_color=BLACK, fill_opacity=0.0, stroke_width=3) + square.move_to(np.array([3.1, -1.5, 0.0])) + self.play(ShowCreation(square), run_time=1) + leaf_points = [ + self.points[5], + self.points[6] + ] + d_node_leaf_left.children = leaf_points + point_group = VGroup(*[point.dot.copy() for point in leaf_points]) + self.play(point_group.animate.move_to(square.get_center()).scale(0.2), run_time=1) + self._draw_arrow(d_graph_node, square) + d_node_leaf_left.graph_node = VGroup(square, point_group) + d_node.children.append(d_node_leaf_left) + + # D leaf node right + d_node_leaf_right = KDTreeNode("D-Right", leaf=True) + square = Square(side_length=0.6, stroke_color=BLACK, fill_opacity=0.0, stroke_width=3) + square.move_to(np.array([3.9, -1.5, 0.0])) + self.play(ShowCreation(square), run_time=1) + leaf_points = [ + self.points[8], + self.points[10], + self.points[12] + ] + d_node_leaf_right.children = leaf_points + point_group = VGroup(*[point.dot.copy() for point in leaf_points]) + self.play(point_group.animate.move_to(square.get_center()).scale(0.3), run_time=1) + self._draw_arrow(d_graph_node, square) + d_node_leaf_right.graph_node = VGroup(square, point_group) + d_node.children.append(d_node_leaf_right) + + # E leaf node left + e_node_leaf_left = KDTreeNode("E-Left", leaf=True) + square = Square(side_length=0.6, stroke_color=BLACK, fill_opacity=0.0, stroke_width=3) + square.move_to(np.array([4.7, -1.5, 0.0])) + self.play(ShowCreation(square), run_time=1) + leaf_points = [ + self.points[7], + self.points[9] + ] + e_node_leaf_left.children = leaf_points + point_group = VGroup(*[point.dot.copy() for point in leaf_points]) + self.play(point_group.animate.move_to(square.get_center()).scale(0.3), run_time=1) + self._draw_arrow(e_graph_node, square) + e_node_leaf_left.graph_node = VGroup(square, point_group) + e_node.children.append(e_node_leaf_left) + + # E leaf node right + e_node_leaf_right = KDTreeNode("E-Right", leaf=True) + square = Square(side_length=0.6, stroke_color=BLACK, fill_opacity=0.0, stroke_width=3) + square.move_to(np.array([5.5, -1.5, 0.0])) + self.play(ShowCreation(square), run_time=1) + leaf_points = [ + self.points[11], + self.points[13] + ] + e_node_leaf_right.children = leaf_points + point_group = VGroup(*[point.dot.copy() for point in leaf_points]) + self.play(point_group.animate.move_to(square.get_center()).scale(0.3), run_time=1) + self._draw_arrow(e_graph_node, square) + e_node_leaf_right.graph_node = VGroup(square, point_group) + e_node.children.append(e_node_leaf_right) + self._delete_info() + return a_node + + + def _draw_arrow(self, n1, n2): + arrow = Arrow( + start=n1.get_bottom(), + end=n2.get_top(), + buff=0, # No spacing at start/end + stroke_width=1, + fill_color=BLACK, + ) + self.play(FadeIn(arrow), run_time=1.0) + + def _draw_graph_node(self, p, label, dim, value): + graph_node = Dot(point=np.array([p[0], p[1], 0.0]), fill_color=WHITE, stroke_color=BLACK, stroke_width=3, radius=0.3) + label = Text(label, font_size=24, fill_color=BLACK) + label.move_to(graph_node.get_center()) + dim = "x" if dim == 0 else "y" + line1 = Text(f"dim: {dim}", font_size=20, fill_color=BLACK) + line1.set_color_by_text("x", BLUE) + line1.set_color_by_text("y", RED) + line2 = Text(f"value: {value}", font_size=20, fill_color=BLACK) + # Stack them vertically with a small gap + val_label = VGroup(line1, line2).arrange(DOWN, buff=0.2) + val_label.move_to(graph_node.get_center()) + val_label.shift(RIGHT*0.8) + self.play(FadeIn(graph_node), FadeIn(label), FadeIn(val_label), run_time=0.5) + return graph_node + + def _draw_line(self, p1, p2, label, label_shift): + line = Line(self.axes.c2p(p1[0], p1[1]), self.axes.c2p(p2[0], p2[1]), color=BLACK, stroke_width=3) + # Add label to the line + label = Text(label, font_size=24, fill_color=BLACK) + label.move_to(line.get_center()) + label.shift(label_shift) + self.play(ShowCreation(line), FadeIn(label), run_time=1.0) + return VGroup(line, label) + + def _title_screen(self, text): + # Create title text + title = Text(text, font_size=72, fill_color=BLACK) + white_screen = Rectangle( + width=FRAME_WIDTH, + height=FRAME_HEIGHT, + color=WHITE, + fill_opacity=1, + stroke_width=0 + ) + + # Center it + white_screen.move_to(ORIGIN) + title.move_to(ORIGIN) + white_screen.set_z_index(20) + title.set_z_index(21) + + # Fade in + self.play(FadeIn(white_screen), FadeIn(title), run_time=1) + # Hold + self.wait(1) + # Fade out + self.play(FadeOut(white_screen), FadeOut(title), run_time=1) + + def _write_info(self, text, coloring): + # Create a Text object with the given text + words = text.split() + lines = [] + current_line = "" + max_line_length = 100 # Maximum length of each line + + for word in words: + if len(current_line) + len(word) + 1 <= max_line_length: + current_line += word + " " + else: + lines.append(current_line.rstrip()) + current_line = word + " " + + if current_line: + lines.append(current_line.rstrip()) + + # Join lines with LaTeX line break + final_text = "\n".join(lines) + text_obj = Text( + final_text, + font="Arial", + font_size=27, + ) + text_obj.set_color(BLACK) + text_obj.move_to(DOWN*3.3 + LEFT*6, aligned_edge=LEFT) + for word, color in coloring.items(): + text_obj.set_color_by_text(word, color) + + # text_obj.arrange(DOWN, buff=0.5) # buff controls spacing + self.text_obj = text_obj + # Add the text to the scene + for letter in self.text_obj: + self.play(FadeIn(letter), run_time=0.05) + + def _delete_info(self): + 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zAL`hT-$7zG+&kP$;aNs*Ht5^amVf=metZwo`S2I||AtHfbL<>Bc5nBz??Iryw%f6O z5mKMz96xgO(X-@yV#v7+`HPaaHIkFmS%{DEaL%=7yoZ2GfiZh-_3K+O0=N$D!)LJ` zw4IX=jMw^gU-wxD#(@67T>S}*XPvcV?VJFHf)zkJ-bd!Y4zdt*1H-`AU_S7!a-Z{o z_N>d=$SL43;9A?`fc}F}#|5;F%NU*o-WBfkzrgv{B{9T}k=_5Jhu^a?-q!(s2VaRT z)beC>ebdf7pkMer>IueeTUTj*U!eUmz`wBfH%P~=m79|GQOHAqxvN)HJYo{kKRKXf zwA1?oki(}I6c0xZm@qO(n$g~43JQ%O{FzWNcFORRPnNq4f0~UgEGkCZ_txS`8f{oG XJS^s5;L-4DquRA<(Y{5imM#Ae)m{j& literal 0 HcmV?d00001 From 30b53e463d24fffc0e94768ef0e16f6f1e0aabf6 Mon Sep 17 00:00:00 2001 From: sujaygarlanka Date: Sat, 10 May 2025 20:46:40 -0400 Subject: [PATCH 2/2] Minor change --- animations/code/motion_planning/kd_tree.py | 2 +- animations/media/motion_planning/KDTree.mp4 | Bin 1683603 -> 1675203 bytes 2 files changed, 1 insertion(+), 1 deletion(-) diff --git a/animations/code/motion_planning/kd_tree.py b/animations/code/motion_planning/kd_tree.py index 9cebc24..4b86eef 100644 --- a/animations/code/motion_planning/kd_tree.py +++ b/animations/code/motion_planning/kd_tree.py @@ -466,7 +466,7 @@ def _draw_graph_node(self, p, label, dim, value): val_label = VGroup(line1, line2).arrange(DOWN, buff=0.2) val_label.move_to(graph_node.get_center()) val_label.shift(RIGHT*0.8) - self.play(FadeIn(graph_node), FadeIn(label), FadeIn(val_label), run_time=0.5) + self.play(FadeIn(graph_node), FadeIn(label), FadeIn(val_label), run_time=1.0) return graph_node def _draw_line(self, p1, p2, label, label_shift): diff --git a/animations/media/motion_planning/KDTree.mp4 b/animations/media/motion_planning/KDTree.mp4 index 9a0d1781499e931283822b588cb3cb101300163a..5676b6309f097eaa4fe2e507360117618731a0a8 100644 GIT binary patch delta 1061504 zcmZ^~1yEc~^XR>}ySo$I-QC^YJ-CHn2X}Y(;O-FICAb6#PH-m>96p}segF5?y>-8; zt*xo8(>;A=x_{F%J)68^HJEReKnI@I9mo*M0M^0)4pj_*B1N;=WXnbEGZkQ+-rveX zK%hT225fqkMl$ct5zDlRb+3h+0Dd4o-i{;)GIWuBbWr^<-G z^|i_MCAO5#>>`NNQ?`omv+5X@CjMyonvSBX+%+T&^S66%Np1q}$rr~WLKoI}) 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