1. The document provides examples and explanations of concepts in solid geometry including the three dimensional coordinate system, distance formula in three space, and equations for planes, spheres, cylinders, quadric surfaces, and their graphs. 2. Key solid geometry concepts covered include plotting points in three dimensions, finding distances between points and distances from a point to a plane, midpoint formulas, and standard and general equations for planes, spheres, cylinders, ellipsoids, hyperboloids, and paraboloids. 3. Examples are given for graphing equations of a plane, sphere, circular cylinder, parabolic cylinder, and their relation to the standard equations.

1. LECTURE UNIT 008
SOLID GEOMETRY
Three Dimensional Coordinate System
z
(0, 0, c)
y
0
(0, b, 0)
(a, 0, 0)
Cartesian coordinates (x, y, z)
x
Plot: P (2, -3, 4)
z
P (2, -3, 4)
4
3
2 y
0
1 1
2
-3 -2 -1
x
Distance Formula in Three Space
z
P1 (x1, y1, x1)
P2 (x2, y2, x2) y
0
x
|P1P2| = d = (x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2
Distance Between a Point (x, y, z) and a plane
Ax1 + By1 + Cz1 + D
d=
A 2 + B2 + C2
Midpoint Formula
x1 + x 2
x=
2
y1 + y2
y=
2
z 1 + z2
z=
2
“Man was designed for accomplishment, engineered for success and endowed with the
seeds of greatness.”

2. Plane
The graph of Ax + By + Cz = D is a plane.
1. 2x + y + 3z = 6
2. 2x + 3y = 6
Sphere
(x - h)2 + (y - k)2 + (z - l)2 = r2 Standard equation of a sphere
x2 + y2 + z2 + Dx + Ey + Fz + G = 0 General equation of a sphere
r2 > 0 the graph is sphere
r2 = 0 the graph is a single point
r2 < 0 no graph
3. x2 + y2 + z2 - 2x + 4y - 6z - 2 = 0
4. 3x2 + 3y2 + 3z2 - 5x + y - 2z + 45 = 0
Cylinders
Any equation in two variables, represent a cylindrical surface, that is perpendicular to the two variables and whose
generating curve is the plane curve whose equation is given. Kinds: Circular, Parabolic, Elliptic or Hyperbolic.
5. x2 + z2 = 4
6. y2 = 4x
7. z2 + 4y2 = 16
2 2 2
8. Sketch the surface whose equation is x + y + z = 1
16 9 4
Quadric Surfaces
Ellipsoid
The surface represented by:
x 2 + y 2 + z2 = 1
a2 b2 c2
is an . Ellipsoid
Hyperboloid of One Sheet
The surface represented by:
x 2 + y 2 - z2 = 1
a2 b2 c2
is a .
Hyperboloid of One Sheet
Hyperboloid of Two Sheets
The surface represented by:
x 2 - y 2 - z2 = 1
a2 b2 c2
is a .
Hyperboloid of Two Sheets
Elliptic Paraboloid
The surface represented by:
x2 - y2 = z
a2 b2
is an .
Elliptic Paraboloid
“Action is the manifestation of learning. Just as “faith without works is dead,” learning without
action isn't learning.”

3. ROTATION OF AXES
The equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 is a second degree equation.
B2 - 4AC is a discriminant of the given equation
if B2 - 4AC = 0 parabola
B2 - 4AC > 0 hyperbola
B2 - 4AC < ellipse
Rotation Formula:
x = x’ cosθ - y’ sinθ
y = x’ sinθ + y’ cosθ
To determine the angle of rotation, we use:
tan2θ = B
A-C
2 tanθ
= B
1 - tan2θ A-C
1. Sketch the graph: 24xy - 7y2 = 144 Original equation
1 A = 0, B = 24, C = -7
2 tan2θ = 24 = 24
0 - (-7) 7
o
2θ = 73.74
θ = 36.87o (angle of rotation)
2 tanθ
3 = 24
1 - tan2θ 7
12 tan2θ + 7 tanθ - 12 = 0
(4 tanθ -3)(3 tanθ + 4) = 0
3 4
tanθ = tanθ =- Disregard negative value of tanθ
4 3
5
3
θ
4
We get;
sinθ =
3 and cosθ = 4
5 5
Then, substitute sinθ and cosθ to rotation formula:
4x’ - 3y’ 3x’ + 4y’
x= y=
5 5
4 Substitute x and y to the original equation, and simplifying, we obtain:
x2' y2' y
- =1 Equation of a hyperbola
16 9
y
x
o
36. 87
3
36. 87o
4
x
3
4
“You measure the size of accomplishment by the obstacles you have to overcome to
reach your goals.”

4. SOLID GEOMETRY
Example 1: Graph 2x + y + 3z = 6 z
x-intercept, set y and z to zero.
x = 3 (3, 0, 0)
y-intercept, set x and z to zero.
y = 6 (0, 6, 0) 2
z-intercept, set x and y to zero. 1
z = 2 (0, 0, 2) 0 1 2 3 4 5 6 y
1
Traces: 2
xy trace set z to zero 3
2x + y = 6 LINE
xz trace set y to zero x
2x + 3z = 6 LINE
yz trace set x to zero z
y + 3z = 6 LINE
Example 2: Graph 2x + 3y = 6
x-intercept,
2
x=3 (3, 0, 0)
1
y-intercept,
0 y
y=2 (0, 2, 0) 1
1 2 3 4 5 6
The plane never crosses the z-axis and so 2
the plane is parallel to the z-axis. 3
z
Example 3: Graph x2 + y2 + z2 - 2x + 4y - 6z - 2 = 0 x
(x2 - 2x + 1) + (y2 + 4y + 4) + (z2 - 6z + 9) = 2 + 1 + 4 + 9
2
(x -1)2 + (y + 2)2 + (z - 3)2 = 16 1
C (1, -2, 3) 0 1 2 3 4 5 6
y
r=4 1
2
3
z
Example 4: Graph x2 + z2 = 4 x
A circular cylinder
C (0, 0) 2
r=2 1
0 1 2 3 4 5 6
y
1
2
3
x z
Example 5: Graph y2 = 4x
A parabolic cylinder
x
“If we want something badly enough, we must make it our definite goal. When we go
after it as if we can't fail, many things will happen to help make certain we won't.”