MATH 251 – LECTURE 5
JENS FORSGÅRD
http://www.math.tamu.edu/~jensf/
This week: 11.3–6
webAssign: 11.3–5, due 2/1 11:55 p.m.
Next week: 11.6–7, 12.1–3
webAssign: 11.6, 12.1, and 12.3, opens 2/1 12 a.m.
Help Sessions:
M W 5.30–8 p.m. in BLOC 161
Office Hours:
BLOC 641C
M 12:30–2:30 p.m.
W 2–3 p.m.
or by appointment.
Quadric curves in R2
The general form of a quadratic equation in two variables is
Ax2 + By 2 + Cxy + Dx + Ey + G = 0.
Example 1. Write x2 + y 2 + 2xy + 2x + 2y = 0 in standard form.
Quadric curves in R2
Example 2. Write x2 + 2xy + 2x + 2y = 0 in standard form.
Quadric curves in R2
Each quadratic equation in two variables can be written in one of the two forms
Ax2 + By 2 + G = 0 and Ax2 + By + G = 0.
Ellipse
Hyperbola
Parabola
ax2 + by 2 − g = 0
ax2 − by 2 ± g = 0
ax2 ± ey ± g = 0
Quadric surfaces in R3
The general form of a quadratic equation in three variables is
Ax2 + By 2 + Cz 2 + Dxy + Exz + F yz + Gx + Hy + Iz + J = 0.
Each quadratic equation in three variables can be written in one of the two forms
Ax2 + By 2 + Cz 2 + J = 0 an Ax2 + By 2 + Iz = 0.
Intercepts:
Traces:
Ellipsoids
ax2 + by 2 + cz 2 = j
One-sheeted hyperboloids
ax2 + by 2 − cz 2 = j,
j>0
Two-sheeted hyperboloids
−ax2 − by 2 + cz 2 = j,
j>0
Cones
ax2 + by 2 − cz 2 = 0
Elliptic paraboloids
ax2 + by 2 ± iz = 0
Hyperbolic paraboloids
ax2 − by 2 ± iz = 0
Quadric surfaces in R3
One-sheeted hyperboloid
Cone
Two-sheeted hyperboloid
-2
-1
2
1
-1
0
-2
2
0
2
1
2
1
2
1
1
0
0
-2
0
-1
-1
-1
-1
-2
0
-2
-2
-2
-1
-2
1
-1
0
1
0
1
2
2
Ellipsoid
-2
-1
0
-2
1
2
2
Elliptic paraboloid
Hyperbolic paraboloid
-1
0
1
2
2
1
4
2.0
2
1.5
0
1.0
0
0.5
-2
-1
1.0
0.0
-1.0
0.5
-0.5
-2
-2
-1
0.0
0.0
-0.5
0.5
2
-2
0
0
-2
2
0
1
2
4
-4
-4
1.0
-1.0
4
-4
Exercises
Exercise 3. Find the traces of the surface x2 − y 2 + 2z 2 = 1. Sketch and identify the surface.
Exercises
Exercise 4. Find the traces of the surface x2 − y 2 − 2z = 0. Sketch and identify the surface.
Exercises
Exercise 5. Find the traces of the surface x2 − y 2 + 2z 2 = 0. Sketch and identify the surface.