Parametric Equations and a Heart
Sometimes the easiest way to create a graph is to use two equations or functions ... an equation for X and
another equation for Y. These equations are often in terms of a separate variable like time or angle size.
These types of equations are called parametric equations. Each value for X and Y are determined by
separate functions that involve a third value or parameter.
The prettiest heart that I found to graph for Valentine's Day uses parametric equations. So let's see what we
can create.
I found these parametric equations for a heart at Wolfram Math World.
Parametric heart
𝑋 = 16 ∙ (𝑠𝑖𝑛 (𝜃))!
𝑌 = 13 ∙ 𝑐𝑜𝑠(𝜃) − 5 ∙ 𝑐𝑜𝑠(2𝜃) − 2 ∙ 𝑐𝑜𝑠(3𝜃) − 𝑐𝑜𝑠(4𝜃)
where −𝜋 < 𝜃 < 𝜋
Let's see how much of these calculations you can do without using your calculator.
1.
Using our slightly distorted image on the left, calculate the sine and
cosine of zero radians and then the parametric values of X and Y.
a. Sin (0) = ________
b. Cos (0) = ________
c.
X = ________
d.
Y = ________________
2. Using the values that you just calculated, mark your first evaluation for (X,Y) on our graph on the second
page of this activity.
3. Using our next slightly distorted image on the right, calculate the sine and
!
cosine of radians and the value of X and Y. Add this point to your graph.
!
!
a. Sin ( ) = ________
!
!
b. Cos ( ) = ________
!
c. X = ________
d. Y = ________________
2𝜋
10
5
0
-5
-10
-15
-20
-20
-15
-10
-5
0
5
10
15
20
4. Find the (X, Y) pairs for the angles pictured above and graph them above.
a. 𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
i. sin 𝜋 = ________
ii. cos 𝜋 = ________
iii. X = ________
iv. Y = ________________
b.
!!
!
𝑟𝑎𝑑𝑖𝑎𝑛𝑠
i. sin
!!
!
!!
=________
ii. cos
= ________
!
iii. X = ________
iv. Y = ________________
All of these painful computations only resulted in 4 points on our graph.
5. What suggestions do you have for attaining more data points for, hopefully, a full and beautiful heart?
Graphing calculator
1. What option do you expect that you will have to enter into your MODE settings in order to create this
graph?
a. Angle types?
b. Function type?
c. Will your graph be connected dots or just dots?
2. From the looks of what we've graphed so far, how should you set your WINDOW ranges?
3. Enter your equations for X and Y and create the graph on your calculator.
4. Are you happy with how your Valentine Heart turned out?
5. If the answer to #9 was no, how can you correct your picture?
EXCEL
So, I've decided to try to enter the data for this heart in a spreadsheet, have Excel calculate all of the
values, and then create a chart of the (X,Y) points.
!
1. I wonder if letting the parameter, Theta, change by ∙ 𝜋 radians for each increment will be enough to
!
make a smooth, good-looking heart. What is your guess?
2. In Excel create a Workbook with 3 columns. We will have a column for 𝜃 measurements, a column for
the calculated X value and a column for the calculated Y value.
!
I'm going to first try increasing the values of 𝜃 by only for each increment = 30o.
!
3. In your first column list all of your 𝜃 values. In your second and third columns (B and C columns), type
the parametric formulas for X and Y. The formulas are a little laborious to type but you only have to
create them once. Then you can fill in the rest of your rows by copying and pasting. Go for it!
A
B
C
1
𝜃 in radians
X
Y
2
0
= 16 ∙ (𝑠𝑖𝑛 (𝐴2))!
= 13 ∙ 𝑐𝑜𝑠(𝐴2) − 5 ∙ 𝑐𝑜𝑠(2 ∗ 𝐴2) − 2 ∙ 𝑐𝑜𝑠(3 ∗ 𝐴2) − 𝑐𝑜𝑠(4 ∗ 𝐴2)
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
!
!
!
!
!
!
!
!
!
!
!
!
!
!
∙𝜋
∙𝜋
∙𝜋
∙𝜋
∙𝜋
∙𝜋
∙𝜋
𝜋
!
−!
!
−!
!
−!
!
−!
!
−!
!
−!
!
−!
∙𝜋
∙𝜋
∙𝜋
∙𝜋
∙𝜋
∙𝜋
∙𝜋
0
Once your data is complete, make a connected scatterplot from just the X and Y columns.
4. Did you have to change your axis scale to create a better-looking heart? Please explain.
Sources: http://mathworld.wolfram.com/HeartCurve.html
http://graphsketch.com/parametric.php
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