More Graphing – manim Series: Part 8

The post is part of a series on learning how to use manim.  You can find the previous tutorial post in this series here and the overview of the entire series here.

Important Note:  These posts are based on an earlier version of manim which uses Python 2.7.  The latest version of manim is using Python 3.  To follow along with these posts, use Python 2.7 and the May 9, 2018 commit of manim .

8.0 More Graphing

In the previous post in this series we looked at how to graph functions in manim so go take a look at this if you haven’t already. I’ve been playing a bit more with graphing in manim and wanted to share some of what I’ve learned. You can copy and paste the code below or download the code from Github.

class ExampleApproximation(GraphScene):
CONFIG = {
"function" : lambda x : np.cos(x),
"function_color" : BLUE,
"taylor" : [lambda x: 1, lambda x: 1-x**2/2, lambda x: 1-x**2/math.factorial(2)+x**4/math.factorial(4), lambda x: 1-x**2/2+x**4/math.factorial(4)-x**6/math.factorial(6),
lambda x: 1-x**2/math.factorial(2)+x**4/math.factorial(4)-x**6/math.factorial(6)+x**8/math.factorial(8), lambda x: 1-x**2/math.factorial(2)+x**4/math.factorial(4)-x**6/math.factorial(6)+x**8/math.factorial(8) - x**10/math.factorial(10)],
"center_point" : 0,
"approximation_color" : GREEN,
"x_min" : -10,
"x_max" : 10,
"y_min" : -1,
"y_max" : 1,
"graph_origin" : ORIGIN ,
"x_labeled_nums" :range(-10,12,2),

}
def construct(self):
self.setup_axes(animate=True)
func_graph = self.get_graph(
self.function,
self.function_color,
)
approx_graphs = [
self.get_graph(
f,
self.approximation_color
)
for f in self.taylor
]

term_num = [
TexMobject("n = " + str(n),aligned_edge=TOP)
for n in range(0,8)]
[t.to_edge(BOTTOM,buff=SMALL_BUFF) for t in term_num]

term = TexMobject("")
term.to_edge(BOTTOM,buff=SMALL_BUFF)

approx_graph = VectorizedPoint(
self.input_to_graph_point(self.center_point, func_graph)
)

self.play(
ShowCreation(func_graph),
)
for n,graph in enumerate(approx_graphs):
self.play(
Transform(approx_graph, graph, run_time = 2),
Transform(term,term_num[n])
)
self.wait()

I wanted to demonstrate how adding higher terms in a Taylor expansion results in better and better agreement with a function. This is similar to what
shinigamiphoenix posted here.

The functions to plot are defined as lambda functions in the CONFIG{} dictionary. manim processes all elements in CONFIG{} and turns the dictionary entries into class variables with the key as the variable name. Thus "function" can be accessed within my class by calling self.function and "taylor" can be called with self.taylor. If you aren’t familiar with lambda functions, check out this post at Python Conquers the Universe.

We create a list of graphs using get_graph() and a list comprehension. You can find a nice tutorial on list comprehensions over at datacamp.com. It was only after reading this tutorial that I made the connection between list comprehensions and mathematical notation for definitions of sets (e.g. the set of positive real numbers is{  R and } or the set of even numbers which is {  I and }), which made list comprehensions click for me. For each item in the list self.taylor, a graph is created with color self.approximation_color. We also created a list of TexMobjects to indicate which order of terms are included from the Taylor expansion using a list comprehension.

Since we are going to do successive transformations from a list, it helps to have a blank placeholder on the screen. term and approx_graph are VectorizedPoint instances, which are mobjects that don’t display anything on screen. This way we can put the placeholders on the screen without anything appearing, and then transform those mobjects into either the graph or the TexMobjects.

The enumerate() command is a useful tool that iterates over a list and also returns the index of the item returned. Thus for n,graph in enumerate(approx_graphs) returns the index between 0 and 4 as n, and the element within the list as graph. This is used to display the corresponding item from term_num with each graph.

 

Check out how to draw vector fields in the next post in this series.