Parametric equations represent an alternative to the Cartesian Coordinate System for graphing curves.
For some families of curves, especially those with circular shapes, graphing them can be made much easier using parametric equations. Often, as is the case with curves with circular shapes, this is because they do not pass the Vertical Line Test, and thus cannot be represented as functions by rectangular equations (i.e. equations in which $y$ is a function of $x$, or vice-versa) without restricting the domain.
Parametric equations eliminate that problem by describing curves as two independent functions of a single, common parameter.
Frequently, the parameter used, by convention, is $t$. Also frequently, the parameter represents time.
e.g. $$x(t)=\cos(t) \quad \quad y(t)=\sin(t)$$ $$0 \le t \le 2\pi$$
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import plotly.plotly as py
import plotly.graph_objs as go
from plotly import offline
import numpy as np
# Configure Plotly to work locally, rather than connecting
# to remote Plotly data (which I don't yet have)
offline.init_notebook_mode()
# Generate the t parameter
t = np.linspace(-8*np.pi, 8*np.pi, 100)
# Radius of the circle
r = np.pi
# Parametric functions
x = r * np.cos(t)
y = r * np.sin(t)
circle = go.Scatter(x=x, y=y)
data = [circle]
layout_config = go.Layout(
width = 700,
height = 700,
title = 'Parametric Circle',
font=dict(
family='Open Sans'
),
titlefont=dict(
size=16,
family='Raleway'
),
xaxis = dict(
title='x(t) = rcos(t)',
gridcolor = 'rgb(245,245,245)',
zerolinecolor = 'rgb(200,200,200)',
#showbackground = True,
),
yaxis=dict(
title='y(t) = rsin(t)',
gridcolor='rgb(245,245,245)',
zerolinecolor='rgb(200,200,200)',
#showbackground=True,
),
)
fig = go.Figure(data=data, layout=layout_config)
offline.iplot(fig)
As the parameter (typically $t$) increases, the parametric curve is generated in a particular direction. This direction is called the forward orientation (sometimes also known as the positive orientation) of the curve.
When $t$ varies between $0 \le t \le 2\pi$, the parametric curve is generated in a counterclockwise orientation.
To obtain parametric equations for a circle with clockwise orientation, we replace $t$ with $-t$, and use the identities $\cos(-t) = \cos(t)$ and $\sin(-t) = -\sin(t)$.
Another way to generate clockwise circles (counterclockwise is the standard/default orientation) is to switch $\cos$ and $\sin$:
$$\boldsymbol{x}=h+r\boldsymbol{\sin}(t), \quad \boldsymbol{y}=k+r\boldsymbol{\cos}(t)$$Note: According to Professor Teague, there are many other ways of controlling the orientation (direction) of the curve.
Also, Professor Teague requires that, on all assignments pertaining to parametric curves, you must draw arrows on the curve to indicate the orientation. </small>
TODO: Fill this out
Consider the curve given by $x=t^2-2t, \quad y=\sqrt{t}$, for $0 \le t \le 4$
a) Determine the $t$-values where the graph intercepts the x-axis or y-axis, and record them below. Sketch the curve over the interval $0\le t \le 4$, labeling the x- and y-intercepts and the initial and terminal points on the sketch as exact-form ordered pairs. Indicate with arrowheads the direction in which the curve is traced as the parameter increases (this is frequently called the orientation of the graph).
b) Eliminate the parameter $t$ to find a Cartesian equation of the curve.
TODO: Fill in remaining questions, graph, and algebraic computations required to complete Problem 1.
The circle with center $(h,k)$ and radius $r$ is given parametrically by $x=h+r\cos(t), y=k+r\sin(t)$, for $0\le t\le2\pi$. Find parametric equations for the path of a particle that moves along the circle $(x+1)^2+y^2=4$ in the manner described:
a) Twice around in the clockwise direction, beginning at $(-1,2)$:
b) Half-way around the circle in the counterclockwisedirection, beginning at $(1,0)$
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