Download Parametric curves on quadratic surfaces

DAY 12
Parametric curves on quadratic surfaces
Curves on conic sections: Recall that
• for curves on ellipses we use cosine and sine,
• for curves on parabolas we use linear and quadratic functions,
• for curves on hyperbolas we use hyperbolic cosine and hyperbolic sine
Examples:
2
2
= 1 is
• A path traversing the ellipse (x 322) + (y+3)
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x(t) = 2 + 3 cos t,
y(t) =
• A path traversing the parabola x
x(t) =
5 + 4(y
6)2
4(t
• A path traversing the hyperbola
2)2
(x
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x(t) = 2 + 3 cosh t,
3 + 2 sin t.
6)2 = 0 is given by
y(t) = t.
(y+6)2
72
y(t) =
= 1 is
6 + 7 sinh t.
The plan is to use this to construct paths on quadratic surfaces. . .
Curves on ellipsoids:
• Let’s build a curve spiraling from top to bottom of the ellipsoid
✓
◆
✓
◆
x 1 2
y 2 2 ⇣ z ⌘2
+
+
=1
2
2
3
making four full turns around.
• Picture:
• Start with z(t) = t. Since we have
3  z  3 we have
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3  t  3.
12. PARAMETRIC CURVES ON QUADRATIC SURFACES
• The ellipsoid equation becomes
✓
◆
✓
◆
x 1 2
y 2 2
+
=1
2
2
Rewrite as
0
0
12
30
t2
9 t2
=
.
9
9
12
y 2 A
x 1 A
@ q
+@ q
=1
2
2
2 9 9t
2 9 9t
• So we choose
x 1
q
= cos (something),
2
2 9 9t
y 2
q
= sin (something)
2
2 9 9t
Here the “something” controls the amount of rotation.
• At t = 0 we are at the “equator” of the ellipse. By t = 3 we want to have rotated
through angle 4⇡. So the “something” should be 4⇡
3 t.
• We obtain
r
r
✓
◆
✓
◆
9 t2
4⇡
9 t2
4⇡
x(t) = 1 + 2
cos
t
y(t) = 2 + 2
sin
t
z(t) = t
9
3
9
3
for 3  t  3.
• Code:
ellipse = ContourPlot3D[{(x - 1)^2/4 + (y - 2)^2/4 + (z)^2/9 == 1},
{x, -5, 5}, {y, -5, 5}, {z, -5, 5},
Axes -> True, AxesLabel -> {"x", "y", "z"},
LabelStyle -> Large, TicksStyle -> Small, Mesh -> 0];
path = ParametricPlot3D[{1 + 2 Sqrt[(9 - t^2)/9] Cos[4 Pi t/3],
2 + 2 Sqrt[(9 - t^2)/9] Sin[4 Pi t/3], t}, {t, -3, 3},
PlotStyle -> Directive[Thick, Red]];
Show[ellipse, path]
Curves on hyperboloids: We can repeat this same procedure to construct a curve on a
hyperboloid
• Consider the (distorted) hyperboloid of one sheet given by
✓
◆
✓
◆
x 2 2
y 2 2 ⇣ z ⌘2
+
=1
2
2
3
• We can again set z(t) = t so that
✓
◆
✓
◆
✓ ◆2
x 2 2
y 2 2
t
9 + t2
+
=1+
=
2
3
3
9
which we rewrite as
x
p
2
3
2
9 + t2
!2
+
✓
y 2
p
9 + t2
◆2
=1
12. PARAMETRIC CURVES ON QUADRATIC SURFACES
• Thus we get a spiral provided
2p
x(t) = 2 +
9 + t2 cos (↵t)
3
y(t) = 2 +
p
9 + t2 sin (↵t)
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z(t) = t
for any constant ↵. Why not choose ↵ = 2⇡
3 ?
• Here is some code and a picture:
pR = 7;
hyperboloid = ContourPlot3D[{(x - 2)^2/4 + (y - 2)^2/9 - (z)^2/9 == 1},
{x, -pR, pR}, {y, -pR, pR}, {z, -pR, pR},
Axes -> True, AxesLabel -> {"x", "y", "z"},
LabelStyle -> Large, TicksStyle -> Small, Mesh -> 0];
Hpath = ParametricPlot3D[{2 + 2 Sqrt[(9 + t^2)/9] Cos[2 Pi t/3],
2 + 3 Sqrt[(9 + t^2)/9] Sin[2 Pi t/3], t}, {t, -10, 10},
PlotStyle -> Directive[Thick, Red]];
Show[hyperboloid, Hpath]
• What if we want a curve to go up and then down on the hyperboloid? Choose
z(t) = 5 t2 . (Plot z(t))
With this choice, the hyperboloid equation becomes
✓
◆
✓
◆
✓
◆2
x 2 2
y 2 2
5 t2
9 + (5 t2 )2
+
=1+
=
2
3
3
9
So we want
p
2p
x=2+
9 + (5 t2 )2 cos (something)
y = 2 + 9 + (5 t2 )2 sin (something)
3
If we want to keep the radial angle constant, so that the curve just travels up & down,
we choose “something” = constant.
If we want the path to rotate about the hyperboloid, we choose “something” = ↵t for
some constant ↵. Here ↵ determines the speed of rotation.
Here are two pictures: The first has “something” = ⇡3 + ↵t with ↵ > 0 small; the
second with ↵ large.
HOMEWORK PROBLEMS
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Curves on paraboloids: These can be constructed in a similar manner.
Homework problems
(1) Find the equation(s) which describe the intersection of the standard cone x2 + y 2 = z 2
with the plane through the point (1, 1, 0) having normal vector h1, 1, 2i. What type of
conic section is formed? Draw a picture.
(2) Find the equation(s) which describe the intersection of the standard cone x2 + y 2 = z 2
with the plane through the point (1, 1, 0) having normal vector h2, 2, 1i. What type of
conic section is formed? Draw a picture.
(3) Consider the ellipsoid
✓
◆
✓
◆
✓
◆
x 2 2
y+1 2
z 1 2
+
+
= 1.
4
3
2
(a) Make a sketch, including information about the dimensions of the ellipsoid.
(b) Find the Cartesian equation of the curve created by the intersection of the ellipsoid
and the plane z = 2.
(c) Find a parametric path which traverses the curve created by the intersection of the
ellipsoid and the plane z = 2.
(d) Find a parametric equation for a curve which circles the ellipsoid in the x z direction
while also oscillating between y = 3 and y = 1. See the following picture:
(4) Repeat the previous problem for the hyperboloid x2 + (y 4)2 z 2 = 1. In part (c), the
parametric curve can only traverse one part of the curve determined by z = 2.