MTH 254 – Mr. Simonds’ class
There are four basic types of graphs in three‐dimensions. These are: Vector‐valued functions A vector‐valued function is a function with one input variable (called a parameter) where the output is a vector. When graphing vector‐valued functions in two or three dimensions, we generally plot only the point at which the head of the vector lies when the vector’s tail is drawn at the origin. The graph of a vector‐valued function is a curve. When graphing derivative values of vector‐valued functions, we tend to draw actual vectors and we place the tail of the vector at the associated point. Multivariable functions A multivariable function is a function where the input is an ordered pair and the output is a single number. In three‐dimensions, the input is almost always a point in the xy‐plane and the output is a value of z. That is, z = f ( x, y ) . The graph of a multivariable function in three‐dimensions is called a surface. Implicit Equations An implicit equation in three‐dimensions is an equation that includes at least one of the three variables x, y, and z where z has not been stated as a function of x and y. Implicit equations in two variables that do not graph to planes are called cylinders in three‐dimensions. There are several implicit equations whose graphs have names. Many of these are categorized as quadric surfaces. Graphs of the basic quadric surfaces appear in section 9.6 of your text. Parametric Surfaces To generate a parametric surface, the variables x, y, and z are all stated as functions of two parameters (generally u and v). Figure 1: From left to right
G
r ( t ) = cos ( t ) sin ( t ) ,sin ( t ) ,cos ( t )
•
a graph of the vector-valued function
•
a graph of the multivariable function z = 0.8 y sin ( x + y )
•
a graph of the implicit equation z − sin ( y ) = 0
•
a graph of the parametric surface x = u , y = sin ( u ) cos ( v ) , z = sin ( u ) sin ( v )
Graphing in three dimensions: Section 10.1 and 10.5|1
MTH 254 – Mr. Simonds’ class
Sketch, in one‐dimension, the solutions to the equation x = 3 . Sketch, in two‐dimensions, the solutions to the equation x = 3 . Sketch, in three‐dimensions, the solutions to the equation x = 3 . Sketch, in three‐dimensions, the solutions to the equation y 2 + z 2 = 4 . 2|Graphing in three dimensions: Sections 10.1 and 10.5
MTH 254 – Mr. Simonds’ class
Sketch, in three‐dimensions, the solutions to the equation y = x 2 . Sketch, in three‐dimensions, the solutions to the equation z = 2 x . Graphing in three dimensions: Section 10.1 and 10.5|3
MTH 254 – Mr. Simonds’ class
G
Consider r1 ( t ) = cos ( t ) , sin ( t ) , t . What are the projections of the curve onto the coordinate planes? What are equations for the surfaces in Figure 2? Figure 2
4|Graphing in three dimensions: Sections 10.1 and 10.5
MTH 254 – Mr. Simonds’ class
G
Consider r2 ( t ) = 5sin ( t ) , − 13 cos ( t ) ,12 sin ( t ) . What are the projections of the curve onto the coordinate planes? What are equations for the surfaces in Figure 3? Figure 3
Graphing in three dimensions: Section 10.1 and 10.5|5
MTH 254 – Mr. Simonds’ class
G
Consider r3 ( t ) = sin ( 2 t ) , cos ( 2 t ) , cos ( 2 t ) . What are the projections of the curve onto the coordinate planes? What are equations for the surfaces in Figure 4? Figure 4
6|Graphing in three dimensions: Sections 10.1 and 10.5
MTH 254 – Mr. Simonds’ class
G
Graph onto Figure 5 the vector‐valued function r4 ( t ) =
t sin ( t ) t cos ( t )
after first completing Table ,
π
π
1. Check your graph with your graphing calculator. Table 1: rG
4
G
r4 ( t )
t
0
π
2
π
3π
2
2π
5π
2
G
Figure 5: r4
3π
G
Graph onto Figure 6 the vector‐valued function r5 ( t ) = 3cos 2 ( t ) , 2sin ( t ) after first completing Table 2. Check your graph with your graphing calculator. Table 2: rG
5
G
t
r4 ( t )
0
π
2
π
3π
2
2π
5π
2
G
Figure 6: r5
3
π
Graphing in three dimensions: Section 10.1 and 10.5|7
MTH 254 – Mr. Simonds’ class
Match each vector function to one of the surfaces shown in figures A – H (page 11). In each case the blue trace curve corresponds to u = 1 and the red trace curve corresponds to v = 1 G
G
r6 ( u, v ) = u , v, 4 − u 2 − v 2 r7 ( u, v ) =
( 3 + cos ( u ) ) cos ( v ) , ( 3 + cos ( u ) ) sin ( v ) , v + sin ( u )
G
G
r8 ( u , v ) = sin 2 ( u ) + cos ( v ) ,cos ( u ) + sin ( v ) ,sin ( u ) r9 ( u, v ) = u cos ( v ) , u sin ( v ) , u G
r10 ( u , v ) = sin ( u ) cos ( v ) , sin ( u ) sin ( v ) , cos ( u ) G
r11 ( u , v ) = u, v 2 ,4 − u 2 − v 2 8|Graphing in three dimensions: Sections 10.1 and 10.5
MTH 254 – Mr. Simonds’ class
While parametric equations with a single parameter can be used to describe curves in two or three dimensions, parametric equations with two parameters can be used to describe surfaces in three dimensions. Let’s describe each of the following surfaces using parametric equations with two parameters. z2
= 1 4
•
the right elliptical cylinder x 2 +
•
the plane that passes through the point Q : ( 2, 9, − 3) that is parallel to the vectors G
G
a = 2, 0, − 1 and b = −2,3,2 Graphing in three dimensions: Section 10.1 and 10.5|9
MTH 254 – Mr. Simonds’ class
•
z x2 y 2
+
the elliptic paraboloid =
9 4
9
10 | G r a p h i n g i n t h r e e d i m e n s i o n s : S e c t i o n s 1 0 . 1 a n d 1 0 . 5