Instructor Hentzel
Office Phone: 515-294-8141
E-mail:   hentzel@iastate.edu
Math Department Fax:      515-294-5454
http://www.math.iastate.edu/hentzel/class.166.ICN

Textbook:  Calculus Early Vectors Preliminary Edition
           By Stewart.


Monday April 7     11.5 Quadric Surfaces 

p687: 1,2,3,9,12,13,(14),17,(18),19,(20),(22),23,(24),25,28,(32),33

Main Idea: Everything is determined by the minus signs on the squared factors. 

Key Words: 
              Quadratic surfaces, 
              sphere, 
              ellipsoid, 
              paraboloid, 
              hyperboloid of one sheet, 
              hyperboloid of two sheets, 
              cone, 
              hyperbolic paraboloid, 
              cylinder
              general quadratic equation of second degree in two variables, 
              rotation,
              translation, 
              trace.
Goal:  Learn recognize the surfaces associated with the equation. 

Ax^2 + By^2 + Cz^2 + Dx + Ey + Fz + G = 0.


Review of quadratics in two variables.

Circles, Ellipses, Hyperbolas, Parabolas.

Standard form:

            x^2       y^2        z^2
          ------ + -------- +  ---------  = 1
            a^2       b^2        c^2

There are 8 possibilities here, and then you have the rest which

have no quadratic terms.

           x^2       y^2      z
          -----  +  ------ = --- 
           a^2       b^2      c

And the number of possibilities is seemingly endless if you try to

include all other variants like

           x^2       z^2      y
          -----  +  ------ = --- 
           a^2       c^2      b


So instead, think of the four types of surfaces and then orient along

the correct axes using trace arguments.  The four main types are

ellipsoid, hyperboloid, paraboloid, hyperbolic paraboloid.



Identify and sketch   4x^2 - y^2 + 2z^2 +4 = 0

Put in standard form

                y^2    z^2
         -x^2 + --- - ----  = 1
                4       2
Identify and sketch

              x^2 + 2 z^2 -6 x - y + 10 = 0.


              y-1  = (x-3)^2 + 2 z^2

Identify and sketch   (a)   x^2 + y^2 = 1  (b)   x^2 + z^2 = 1



p = y-1 -(x-3)^2 - 2z^2; 
ans = Solve[p==0,z];
f1[x_,y_] = z /. ans[[1]];
f2[x_,y_] = z /. ans[[2]];

a = Plot3D[ f1[x,y],{x,-10,10},{y,-10,10},PlotPoints->100];
b = Plot3D[ f2[x,y],{x,-10,10},{y,-10,10},PlotPoints->100];
c = Show[a,b,PlotLabel->"y-1 = (x-3)^2 + 2 z^2"];
Display["tang.ps",%];