Tag Archives: parametric curve

Drawing General Ellipse with Python Turtle

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We drew an ellipse centered at origin without tilt. In this project, we will draw a general ellipse that can be centered at any location and any tilt angle. The parametric equation for this general ellipse is as follows:

x = cx + a*math.cos(t)*math.cos(math.radians(angle))-b*math.sin(t)*math.sin(math.radians(angle))
y = cy + a*math.cos(t)*math.sin(math.radians(angle))+b*math.sin(t)*math.cos(math.radians(angle))

(cx, cy) is the coordinate of the center of the ellipse. angle is the tilt angle of the ellipse. Range parameter t from 0 to 2π gradually to draw general ellipses.

General Ellipses with Python Turtle

Solution:

import turtle
import math
screen = turtle.Screen()
screen.setup(1000,1000)
screen.title("General Ellipse with Parametric Equation - PythonTurtle.Academy")
screen.tracer(0,0)

turtle.speed(0)
turtle.hideturtle()
turtle.up()

n = 2000
# (cx,cy): center of ellipse, a: width, b:height, angle: tilt
def ellipse(cx,cy,a,b,angle):
    # draw the first point with pen up
    t = 0
    x = cx + a*math.cos(t)*math.cos(math.radians(angle))-b*math.sin(t)*math.sin(math.radians(angle))
    y = cy + a*math.cos(t)*math.sin(math.radians(angle))+b*math.sin(t)*math.cos(math.radians(angle))
    turtle.up()
    turtle.goto(x,y)
    turtle.down()
    turtle.color('red')
    # draw the rest with pen down
    for i in range(n):
        x = cx + a*math.cos(t)*math.cos(math.radians(angle))-b*math.sin(t)*math.sin(math.radians(angle))
        y = cy + a*math.cos(t)*math.sin(math.radians(angle))+b*math.sin(t)*math.cos(math.radians(angle))
        turtle.goto(x,y)    
        t += 2*math.pi/n

ellipse(-200,-100,200,100,45)
ellipse(200,100,40,200,80)
turtle.update()

Drawing Ellipse with Parametric Equation in Python Turtle

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We have several projects drawing ovals. Ellipse is not the same oval. An ellipse is a curve surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve. Ellipse is a generalization of a circle (where two focal points at the same location). The parametric equations for an ellipse is as follows:

x = a*math.cos(t)
y = b*math.sin(t)

The equations are very similar to those of circle except that instead of one single radius, we have two: a, b. Use the equations to draw the following ellipse:

Ellipse with Python Turtle

Drawing Circle with Parametric Equation

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Although there is built in function to draw a circle in Turtle, let’s draw circles in a different way with parametric equation. The parametric equations for a circle is very simple:

x = radius*math.cos(t)
y = radius*math.sin(t)

In the equations above t is the parameter, x, y are the coordinates of the points on the circle. Let t start from 0 and go up to 2π gradually.

Drawing Circle with Parametric Equation with Python Turtle

Lissajous Curve with Python Turtle

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Lissajous Curve is a famous curve with both practical use and in art and design. It is generated by a simple parametric equation:

x = 300*math.cos(k1*t)
y = 300*math.sin(k2*t)

In the equation above t is the parameter. Let it range gradually from 0 to 4π. Set k1 to 3 and k2 to 2 and drawing the following shape. You can play with other numbers for k1 and k2 to see different curves.

Lissajous Curve with Python Turtle