Conic Sections (2D) Cylinders and Quadric Surfacesyinzhang/2011-summer114/july12.pdf · Conic...

Conic Sections (2D)Cylinders and Quadric Surfaces

What you will learn today

Conic Sections (in 2D coordinates)

Cylinders (3D)

Quadric Surfaces (3D)

Vectors and the Geometry of Space 1/24


ParabolasellipsesHyperbolasShifted Conics

Conic sections result from intersecting a cone with a plane.







A parabola is the set of points in a plane that are equidistant froma fixed point F (called the focus) and a fixed line (called thedirectrix). The point halfway between the focus and the directrix ison the parabola, it is called the vertex. The line perpendicular tothe directrix and through the focus is called the axis.




Choose the origin O to be at the vertex, the y-axis as the axis ofthe parabola, the focus F(0,p). For a point P(x,y) on the parabola,have

|PF | =√x2 + (y − p)2 = |y + p| ⇒

x2 = 4py

In general, it opens upward if p¿0 and downward if p¡0. It issymmetric about the y-axis.If we interchange x and y in the equations, y2 = 4px is theparabola with focus (p,0) and directrix x = −p. It opens to theright if p¿0 and opens to the left if p¡0.




Find the focus and directrix of the parabola y2 + 10x = 0 andsketch the graph.Reflection properties of parabola.




An ellipse is the set of points in a plane the sum of whosedistances from two fixed points F1 and F2 is a constant. F1 and F2are called foci.Kepler’s law 1 says that the orbits of the planets in the solarsystem are ellipses with the sun at one focus.Put the foci on the x-axis (−c , 0) and (c, 0), the sum of distance is2a > 0. Therefore the vertices of ellipse are (−a, 0) and (a, 0) onthe x-axis. The line segment between the two vertices is called themajor axis. When c=0, the two foci coincide and the ellipse is acircle.P(x , y) is on the ellipse,

|PF1|+ |PF2| = 2a

Put b2 = a2 − c2, the above equation can be written as

x2

a2+

y2

b2= 1




On the other hand, the ellipse

x2

b2+

y2

a2= 1, a ≥ b > 0

has foci (0,±c) and vertices (0,±a) on the y-axis.




1. Sketch the graph of 9x2 + 16y2 = 144 and locate the foci.2. Find an equation of the ellipse with foci (0,±2) and vertices(0,±3).3. Reflection properties of ellipse.




A hyperbola is the set of all points in a plane the difference ofwhose distances from two fixed points F1 and F2 is a constant.




Put the vertices F1, F2 = (±c , 0) along the x-axis, the vertices are(±a, 0). Have

|PF1| − |PF2| = ±2a

Write b2 = c2 − a2. The equation of the parabola is

x2

a2− y2

b2= 1

A hyperbola has two branches. It has asymptotes

y = ±b

ax




On the other hand, the hyperbola with foci (0,±c), vertices(0,±a) on the y-axis has form

y2

a2− x2

b2= 1

It has asymptotes y = ± abx




If we shift the origin h units to the right and k units up, we willreplace x and y by x-h and y-k in the equations.Sketch the conic 9x2 − 4y2 − 72x + 8y + 176 = 0



Quadric Surfaces

General surface:To Sketch the graph of a surface, it is useful to determine thecurves of intersection if the surface with planes parallel to thecoordinate planes. These curves are called traces (orcross-sections) of the surface.Cylinders:A cylinder is a surface that consists all lines (called rulings) thatare parallel to a given line and pass through a given plane curve.The parabolic cylinder: x = ay2, a > 0



Quadric Surfaces

When one of the variable x, y or z (say x) is missing from theequation, then the surface is a cylinder, and the rulings are parallelto (say x).Example: Sketch x2 + y2 = 1.



Quadric Surfaces

A quadric surface is the graph of a second-degree equation in threevariables x, y and z. The most general such equation is

Ax2 + By2 + Cz2 + Dxy + Eyz + Fxz + Gx + Hy + Iz + J = 0

where the capital letters are constants (some of them could be 0!).By translation and rotation it can be brought into the standardforms

Ax2 + By2 + Cz2 + J = 0

orAx2 + By2 + Iz = 0



Quadric Surfaces

Example: use traces to sketch the surface with equation

x2

4+

y2

9+

z2

4= 1

In general, an ellipsoid has equation

x2

a2+

y2

b2+

z2

c2= 1



Quadric Surfaces

Example:z = 4x2 + y2

This is called an elliptic paraboloid.



Quadric Surfaces

Example:z = y2 − x2

This is called a hyperbolic paraboloid (or saddle),



Quadric Surfaces

Example:x2

4+ y2 − z2

4= 1

Hyperboloid of one sheet,



Quadric Surfaces

Example:4x2 − y2 + 2z2 + 4 = 0

Hyperboloid of two sheets,

whether a hyperboloid is one sheet or two sheets depends onwhether z could be 0 or not.



Quadric Surfaces

Example: Cones:x2

a2+

y2

b2− z2

c2= 0



Quadric Surfaces

Applications:A satellite dish is a paraboloid.Cooling towers in power stations is in the shape of hyperboloids.Twin gears in a triturating juicer,



Quadric Surfaces

What you have learned today

Conic Sections (in 2D coordinates)

Cylinders (3D)

Quadric Surfaces (3D)


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Conic Sections (2D) Cylinders and Quadric Surfacesyinzhang/2011-summer114/july12.pdf · Conic...

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