James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

laboratory project Bézier Curves 657
66. x − e t 2 t, y − 4e ty2 , 0 < t < 1
67. If f 9 is continuous and f 9std ± 0 for a < t < b, show that the
parametric curve x − f std, y − tstd, a < t < b, can be put in
the form y − Fsxd. [Hint: Show that f 21 exists.]
68. Use Formula 1 to derive Formula 6 from Formula 8.2.5 for
the case in which the curve can be represented in the form
y − Fsxd, a < x < b.
69. The curvature at a point P of a curve is defined as
d
− Z
ds
Z
where is the angle of inclination of the tangent line at P, as
shown in the figure. Thus the curvature is the absolute value
of the rate of change of with respect to arc length. It can be
regarded as a measure of the rate of change of direction of the
curve at P and will be studied in greater detail in Chapter 13.
(a) For a parametric curve x − xstd, y − ystd, derive the
formula
− |ẋÿ 2 ẍẏ |
fẋ 2 1 ẏ 2 g 3y2
where the dots indicate derivatives with respect to t, so
ẋ − dxydt. [Hint: Use − tan 21 sdyydxd and Formula 2
to find dydt. Then use the Chain Rule to find dyds.]
(b) By regarding a curve y − f sxd as the parametric curve
x − x, y − f sxd, with parameter x, show that the formula
in part (a) becomes
−
| d 2 yydx 2 |
f1 1 sdyydxd 2 g 3y2
y
70. (a) Use the formula in Exercise 69(b) to find the curvature of
the parabola y − x 2 at the point s1, 1d.
(b) At what point does this parabola have maximum curvature?
71. Use the formula in Exercise 69(a) to find the curvature of the
cycloid x − 2 sin , y − 1 2 cos at the top of one of its
arches.
72. (a) Show that the curvature at each point of a straight line
is − 0.
(b) Show that the curvature at each point of a circle of
radius r is − 1yr.
73. A string is wound around a circle and then unwound while
being held taut. The curve traced by the point P at the end of
the string is called the involute of the circle. If the circle has
radius r and center O and the initial position of P is sr, 0d, and
if the parameter is chosen as in the figure, show that parametric
equations of the involute are
x − rscos 1 sin d
y
O
r
¨
y − rssin 2 cos d
74. A cow is tied to a silo with radius r by a rope just long enough
to reach the opposite side of the silo. Find the grazing area
available for the cow.
T
P
x
P
˙
0 x
laboratory Project
; Bézier curves
Bézier curves are used in computer-aided design and are named after the French mathematician
Pierre Bézier (1910–1999), who worked in the automotive industry. A cubic Bézier curve is
determined by four control points, P 0sx 0, y 0d, P 1sx 1, y 1d, P 2sx 2, y 2d, and P 3sx 3, y 3d, and is defined
by the parametric equations
x − x 0s1 2 td 3 1 3x 1ts1 2 td 2 1 3x 2t 2 s1 2 td 1 x 3t 3
y − y 0s1 2 td 3 1 3y 1ts1 2 td 2 1 3y 2t 2 s1 2 td 1 y 3t 3
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.