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Let $ P $ be a point at a distance $ d $ from the center of a circle of radius $ r $. The curve traced out by $ P $ as the circle rolls along a straight line is called a trochoid. (Think of the motion of a point on a spoke of a bicycle wheel.) The cycloid is the special case of a trochoid with $ d = r $. Using the same parameter $ \theta $ as for the cycloid, and assuming the line is the $ x $-axis and $ \theta = 0 $ when $ P $ is at one of its lowest points, show that parametric equations of the trochoid are$$ x = r\theta - d \sin \theta \quad y = r - d \cos \theta $$ Sketch the trochoid for the cases $ d < r $ and $ d > r $.

Suppose the circle has rotated through radian. Because the circle has been in contact with the line, we see from Figure that the distance it has rolled from the origin is $O T=\operatorname{arc} A T=r \theta$

calculate $\mathrm{X}$$x=O T-P Q$$=r \theta-d \sin \theta$

calculate y$y=C T-C Q$

Graph not available$=r-d \cos \theta$

Calculus 2 / BC

Chapter 10

Parametric Equations and Polar Coordinates

Section 1

Curves Defined by Parametric Equations

Parametric Equations

Polar Coordinates

Missouri State University

Campbell University

University of Michigan - Ann Arbor

Lectures

01:11

In mathematics, integratio…

06:55

In grammar, determiners ar…

14:24

Hypocycloid A circle $C$ o…

05:03

The figure shows a circle …

01:05

Find parametric equations …

02:21

A cycloid is the path trac…

01:55

Let $A$ and $B$ be the poi…

05:42

Hypocycloid any point $P$ …

10:26

Tilted ellipse Consider th…

01:09

Epicycloid A circle of rad…

02:01

As the point $N$ moves alo…

09:54

Longbow Curve In the follo…

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In mathematics, integration is one of the two main operations in calculus, w…

In grammar, determiners are a class of words that are used in front of nouns…

Hypocycloid A circle $C$ of radius $b$ rolls on the inside of a larger circl…

The figure shows a circle of radius $a$ rolling along a horizontal line. Poi…

Find parametric equations for the path traced out by a specific point inside…

A cycloid is the path traced by a point on a rolling circle (think of a ligh…

Let $A$ and $B$ be the points where the ray of angle $\theta$ intersects the…

Hypocycloid any point $P$ on the circumference of the rolling circle describ…

Tilted ellipse Consider the curve $\mathbf{r}(t)=\langle\cos t, \sin t, c \s…

Epicycloid A circle of radius 1 rolls around the outside of a circle of radi…

As the point $N$ moves along the line $y=a$ in the accompanyingfigure, $…

Longbow Curve In the following figure, the circle of radius $a$ is stationar…

03:51

Plot the space curve and its curvature function $ \kappa (t) $. Comment on h…

02:30

The graph of $ r(t) = \langle t - \frac{3}{2} \sin t, 1 - \frac{3}{2} \cos t…

03:13

(a) Sketch the curve by using the parametric equations to plot points. Indic…

03:33

A car is traveling at $ 100\;km/h $ when the driver sees an accident $ 80\;m…

01:59

Find the length of the curve.

$ r(t) = \langle t, 3 \cos t, 3 \sin t…

02:25

Find parametric equations for the tangent line to the curve with the given p…

The swallowtail catastrophe curves are defined by the parametric equations $…

06:32

(a) Approximate $ f $ by a Taylor polynomial with degree $ n $ at the number…

01:26

Compare the curves represented by the parametric equations. How do they diff…

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Find the unit tangent vector $ T(t) $ at the point with the given value of t…

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