3 d coordinate system

1. 3D Coordinate System

2. 3D Coordinate System
To set up the 3D coordinate system, we add
a z–axis which is perpendicular to both the
x&y axes.

3. 3D Coordinate System
To set up the 3D coordinate system, we add
a z–axis which is perpendicular to both the
x&y axes. There are two ways to add the z–axis.

4. 3D Coordinate System
To set up the 3D coordinate system, we add
a z–axis which is perpendicular to both the
x&y axes. There are two ways to add the z–axis.
x
y
z+
Right–hand systemLeft–hand system
x
y
z+

5. 3D Coordinate System
To set up the 3D coordinate system, we add
a z–axis which is perpendicular to both the
x&y axes.
x
y
z+
Right–hand systemLeft–hand system
x
y
z+
Right–hand systemLeft–hand system
There are two ways to add the z–axis.

6. 3D Coordinate System
To set up the 3D coordinate system, we add
a z–axis which is perpendicular to both the
x&y axes.
x
y
z+
Right–hand systemLeft–hand system
x
y
z+
Right–hand systemLeft–hand system
In math/sci, we use the
right–hand system.
The left hand system is
used in computer
graphics for the virtual
space beyond the screen.
There are two ways to add the z–axis.

7. 3D Coordinate System
To set up the 3D coordinate system, we add
a z–axis which is perpendicular to both the
x&y axes.
x
y
z+
Right–hand systemLeft–hand system
x
y
z+
Right–hand systemLeft–hand system
In math/sci, we use the
right–hand system.
The left hand system is
used in computer
graphics for the virtual
space beyond the screen.
We write the 2D plane
and the 3D space
respectively as R2 and R3.
There are two ways to add the z–axis.

8. 3D Coordinate System
Every position in space, may be addressed by
three numbers (x, y, z), called an ordered triple.
x
y
z+

9. 3D Coordinate System
Every position in space, may be addressed by
three numbers (x, y, z), called an ordered triple.
x
y
z+
Given (x, y, z), to find the
location it represents:

10. 3D Coordinate System
Every position in space, may be addressed by
three numbers (x, y, z), called an ordered triple.
x
y
z+
Given (x, y, z), to find the
location it represents:
1. find (x, y) in the x&y
coordinate plane

11. 3D Coordinate System
Every position in space, may be addressed by
three numbers (x, y, z), called an ordered triple.
x
y
z+
Given (x, y, z), to find the
location it represents:
1. find (x, y) in the x&y
coordinate plane
2. the z gives the location
of the point, z units above
or below (x, y).
z > 0  above
z < 0  below

12. 3D Coordinate System
Every position in space, may be addressed by
three numbers (x, y, z), called an ordered triple.
y
z+
x
Ex: Draw A(2, 0 , 0), B(1, 3, 4), C(–2, –1, –3)
Given (x, y, z), to find the
location it represents:
1. find (x, y) in the x&y
coordinate plane
2. the z gives the location
of the point, z units above
or below (x, y).
z > 0  above
z < 0  below

13. 3D Coordinate System
Every position in space, may be addressed by
three numbers (x, y, z), called an ordered triple.
y
z+
Ex: Draw A(2, 0 , 0), B(1, 3, 4), C(–2, –1, –3)
x
A(2, 0 , 0),
Given (x, y, z), to find the
location it represents:
1. find (x, y) in the x&y
coordinate plane
2. the z gives the location
of the point, z units above
or below (x, y).
z > 0  above
z < 0  below

14. 3D Coordinate System
Every position in space, may be addressed by
three numbers (x, y, z), called an ordered triple.
y
z+
Ex: Draw A(2, 0 , 0), B(1, 3, 4), C(–2, –1, –3)
x
A(2, 0 , 0),
(1, 3, 0),
Given (x, y, z), to find the
location it represents:
1. find (x, y) in the x&y
coordinate plane
2. the z gives the location
of the point, z units above
or below (x, y).
z > 0  above
z < 0  below

15. 3D Coordinate System
Every position in space, may be addressed by
three numbers (x, y, z), called an ordered triple.
y
z+
Given (x, y, z), to find the
location it represents:
1. find (x, y) in the x&y
coordinate plane
2. the z gives the location
of the point, z units above
or below (x, y).
z > 0  above
z < 0  below
Ex: Draw A(2, 0 , 0), B(1, 3, 4), C(–2, –1, –3)
x
A(2, 0 , 0),
B(1, 3, 4),
(1, 3, 0),

16. 3D Coordinate System
Every position in space, may be addressed by
three numbers (x, y, z), called an ordered triple.
y
z+
Given (x, y, z), to find the
location it represents:
1. find (x, y) in the x&y
coordinate plane
2. the z gives the location
of the point, z units above
or below (x, y).
z > 0  above
z < 0  below
Ex: Draw A(2, 0 , 0), B(1, 3, 4), C(–2, –1, –3)
x
A(2, 0 , 0),
B(1, 3, 4),
C(–2, –1, –3)
(1, 3, 0),
(–2, –1, 0),

17. 3D Coordinate System
3D coordinate may be drawn from different eye
positions:

18. 3D Coordinate System
3D coordinate may be drawn from different eye
positions:
x
y
z+
Eye:(1, –1, 1)

19. 3D Coordinate System
3D coordinate may be drawn from different eye
positions:
x
y
z+
x
y
z+
Eye:(1, –1, 1) Eye:(1, 1, 1)

20. 3D Coordinate System
3D coordinate may be drawn from different eye
positions:
x
y
z+
x
y
z+
x
y
z+
Eye:(1, –1, 1) Eye:(1, 1, 1) Eye:(1, –1, –1)

21. 3D Coordinate System
3D coordinate may be drawn from different eye
positions:
x
y
z+
x
y
z+
x
y
z+
Eye:(1, –1, 1) Eye:(1, 1, 1) Eye:(1, –1, –1)
There are three coordinate planes:
y
z+
x

22. 3D Coordinate System
3D coordinate may be drawn from different eye
positions:
y
z+
xy–plane
x
y
z+
x
y
z+
x
y
z+
Eye:(1, –1, 1) Eye:(1, 1, 1) Eye:(1, –1, –1)
There are three coordinate planes:
the xy–plane,
x

23. 3D Coordinate System
3D coordinate may be drawn from different eye
positions:
x
y
z+
x
y
z+
x
y
z+
Eye:(1, –1, 1) Eye:(1, 1, 1) Eye:(1, –1, –1)
y
z+
xy–plane
xz–plane
There are three coordinate planes:
the xy–plane,
the xz–plane,
x

24. 3D Coordinate System
3D coordinate may be drawn from different eye
positions:
x
y
z+
x
y
z+
x
y
z+
Eye:(1, –1, 1) Eye:(1, 1, 1) Eye:(1, –1, –1)
There are three coordinate planes:
the xy–plane,
the xz–plane,
and the yz–plane.
x
y
z+
xy–plane
xz–plane
yz–plane
Points in the xy plane are
(x, y, 0), i.e. defined by z = 0.

25. 3D Coordinate System
Some Basic Equations and Their Graphs in 3D

26. 3D Coordinate System
Some Basic Equations and Their Graphs in 3D
In general, the graph of an equation in R3 is a surface.

27. 3D Coordinate System
Some Basic Equations and Their Graphs in 3D
In general, the graph of an equation in R3 is a surface.
The constant equations:
The graphs of the equations
x=k, y=k, or z=k are planes
that are parallel to the coordinate
planes.

28. 3D Coordinate System
Some Basic Equations and Their Graphs in 3D
x
z+
In general, the graph of an equation in R3 is a surface.
The constant equations:
The graphs of the equations
x=k, y=k, or z=k are planes
that are parallel to the coordinate
planes.
Example: a. x = 4 is
a plane // to the yz–plane
y

29. 3D Coordinate System
Some Basic Equations and Their Graphs in 3D
x
z+
In general, the graph of an equation in R3 is a surface.
The constant equations:
The graphs of the equations
x=k, y=k, or z=k are planes
that are parallel to the coordinate
planes.
Example: a. x = 4 is
a plane // to the yz–plane x = 4
y

30. 3D Coordinate System
Some Basic Equations and Their Graphs in 3D
x
z+
In general, the graph of an equation in R3 is a surface.
The constant equations:
The graphs of the equations
x=k, y=k, or z=k are planes
that are parallel to the coordinate
planes.
Example: a. x = 4 is
a plane // to the yz–plane x = 4
b. y = 4 is a plane //
to the xz–plane
y

31. 3D Coordinate System
x
z+
Example: a. x = 4 is
a plane // to the yz–plane x = 4
b. y = 4 is a plane //
to the xz–plane
yy = 4
Some Basic Equations and Their Graphs in 3D
In general, the graph of an equation in R3 is a surface.
The constant equations:
The graphs of the equations
x=k, y=k, or z=k are planes
that are parallel to the coordinate
planes.

32. 3D Coordinate System
x
z+
Example: a. x = 4 is
a plane // to the yz–plane x = 4
b. y = 4 is a plane //
to the xz–plane
yy = 4
c. z = 4 is a plane //
to the xy–plane
Some Basic Equations and Their Graphs in 3D
In general, the graph of an equation in R3 is a surface.
The constant equations:
The graphs of the equations
x=k, y=k, or z=k are planes
that are parallel to the coordinate
planes.

33. 3D Coordinate System
x
z+
Example: a. x = 4 is
a plane // to the yz–plane x = 4
b. y = 4 is a plane //
to the xz–plane
yy = 4
c. z = 4 is a plane //
to the xy–plane
z = 4
Some Basic Equations and Their Graphs in 3D
In general, the graph of an equation in R3 is a surface.
The constant equations:
The graphs of the equations
x=k, y=k, or z=k are planes
that are parallel to the coordinate
planes.

34. 3D Coordinate System
Example: Sketch – y + 2z = 4 in 3D
Some Basic Equations and Their Graphs in 3D

35. 3D Coordinate System
Example: Sketch – y + 2z = 4 in 3D
Set x=y=0  z=2, get (0, 0 ,2)
Some Basic Equations and Their Graphs in 3D

36. 3D Coordinate System
Example: Sketch – y + 2z = 4 in 3D
Set x=y=0  z=2, get (0, 0 ,2)
Set x=z=0  y=–4, get (0, –4, 0)
Some Basic Equations and Their Graphs in 3D

37. 3D Coordinate System
Example: Sketch – y + 2z = 4 in 3D
Set x=y=0  z=2, get (0, 0 ,2)
Set x=z=0  y=–4, get (0, –4, 0)
Set y=z=0, it not possible, so
there is no x intercept.
Some Basic Equations and Their Graphs in 3D

38. 3D Coordinate System
Example: Sketch – y + 2z = 4 in 3D
Set x=y=0  z=2, get (0, 0 ,2)
Set x=z=0  y=–4, get (0, –4, 0)
Set y=z=0, it not possible, so
there is no x intercept.
Some Basic Equations and Their Graphs in 3D
x
z+
y
(0, 0, 2)
(0, –4, 0)

39. 3D Coordinate System
Example: Sketch – y + 2z = 4 in 3D
Set x=y=0  z=2, get (0, 0 ,2)
Set x=z=0  y=–4, get (0, –4, 0)
Set y=z=0, it not possible, so
there is no x intercept.
Plot the y and z intercepts, the line
– y + 2z = 4 in the yz–plane is part
of the graph.
Some Basic Equations and Their Graphs in 3D
x
z+
y
(0, 0, 2)
(0, –4, 0)

40. 3D Coordinate System
Example: Sketch – y + 2z = 4 in 3D
Set x=y=0  z=2, get (0, 0 ,2)
Set x=z=0  y=–4, get (0, –4, 0)
Set y=z=0, it not possible, so
there is no x intercept.
Since the equation doesn't have x, so x may assume any
value.
Plot the y and z intercepts, the line
– y + 2z = 4 in the yz–plane is part
of the graph.
Some Basic Equations and Their Graphs in 3D
x
z+
y
(0, 0, 2)
(0, –4, 0)

41. 3D Coordinate System
Example: Sketch – y + 2z = 4 in 3D
Set x=y=0  z=2, get (0, 0 ,2)
Set x=z=0  y=–4, get (0, –4, 0)
Set y=z=0, it not possible, so
there is no x intercept.
Since the equation doesn't have x, so x may assume any
value. Hence a point (0, a, b) on the line –y + 2z = 4 gives
infinite many solutions (#, a, b) and they form a line parallel to
the x–axis.
Plot the y and z intercepts, the line
– y + 2z = 4 in the yz–plane is part
of the graph.
Some Basic Equations and Their Graphs in 3D
x
z+
y
(0, 0, 2)
(0, –4, 0)

42. 3D Coordinate System
x
z+
Example: Sketch – y + 2z = 4 in 3D
Set x=y=0  z=2, get (0, 0 ,2)
Set x=z=0  y=–4, get (0, –4, 0)
Set y=z=0, it not possible, so
there is no x intercept.
y
Since the equation doesn't have x, so x may assume any
value. Hence a point (0, a, b) on the line –y + 2z = 4 gives
infinite many solutions (#, a, b) and they form a line parallel to
the x–axis.
All such parallel lines passing through –y + 2x = 4 form a
plane, that is parallel to the x–axis, is the graph.
Plot the y and z intercepts, the line
– y + 2z = 4 in the yz–plane is part
of the graph.
Some Basic Equations and Their Graphs in 3D
(0, 0, 2)
(0, –4, 0)

43. 3D Coordinate System
x
z+
Example: Sketch – y + 2z = 4 in 3D
Set x=y=0  z=2, get (0, 0 ,2)
Set x=z=0  y=–4, get (0, –4, 0)
Set y=z=0, it not possible, so
there is no x intercept.
y
Since the equation doesn't have x, so x may assume any
value. Hence a point (0, a, b) on the line –y + 2z = 4 gives
infinite many solutions (#, a, b) and they form a line parallel to
the x–axis.
All such parallel lines passing through –y + 2x = 4 form a
plane, that is parallel to the x–axis, is the graph.
Plot the y and z intercepts, the line
– y + 2z = 4 in the yz–plane is part
of the graph.
Some Basic Equations and Their Graphs in 3D
(0, 0, 2)
(0, –4, 0)

44. 3D Coordinate System
The linear equations:
The graphs of ax + by + cz = d are
planes.
Some Basic Equations and Their Graphs in 3D

45. 3D Coordinate System
The linear equations:
The graphs of ax + by + cz = d are
planes. Use the intercepts to graph:
Set x=y=0 to get the z intercept,
set x=z=0 to get the y intercept,
set y=z=0 to get the x intercept.
Some Basic Equations and Their Graphs in 3D

46. 3D Coordinate System
The linear equations:
The graphs of ax + by + cz = d are
planes. Use the intercepts to graph:
Set x=y=0 to get the z intercept,
set x=z=0 to get the y intercept,
set y=z=0 to get the x intercept.
Example: Sketch 2x – y + 2z = 4
Some Basic Equations and Their Graphs in 3D

47. 3D Coordinate System
The linear equations:
The graphs of ax + by + cz = d are
planes. Use the intercepts to graph:
Set x=y=0 to get the z intercept,
set x=z=0 to get the y intercept,
set y=z=0 to get the x intercept.
Example: Sketch 2x – y + 2z = 4
Set x=y=0  z=2, get (0, 0, 2)
Some Basic Equations and Their Graphs in 3D

48. 3D Coordinate System
The linear equations:
The graphs of ax + by + cz = d are
planes. Use the intercepts to graph:
Set x=y=0 to get the z intercept,
set x=z=0 to get the y intercept,
set y=z=0 to get the x intercept.
Example: Sketch 2x – y + 2z = 4
Set x=y=0  z=2, get (0, 0, 2)
Set x=z=0  y=–4, get (0, –4, 0)
Some Basic Equations and Their Graphs in 3D

49. 3D Coordinate System
The linear equations:
The graphs of ax + by + cz = d are
planes. Use the intercepts to graph:
Set x=y=0 to get the z intercept,
set x=z=0 to get the y intercept,
set y=z=0 to get the x intercept.
Example: Sketch 2x – y + 2z = 4
Set x=y=0  z=2, get (0, 0, 2)
Set x=z=0  y=–4, get (0, –4, 0)
Set y=z=0  x=2, get (2, 0, 0).
Three points determine a plane.
Some Basic Equations and Their Graphs in 3D

50. 3D Coordinate System
x
z+
The linear equations:
The graphs of ax + by + cz = d are
planes. Use the intercepts to graph:
Set x=y=0 to get the z intercept,
set x=z=0 to get the y intercept,
set y=z=0 to get the x intercept.
Example: Sketch 2x – y + 2z = 4
Set x=y=0  z=2, get (0, 0, 2)
Set x=z=0  y=–4, get (0, –4, 0)
Set y=z=0  x=2, get (2, 0, 0).
y
Three points determine a plane. Plot these three
intercepts and the plane containing them is the graph.
Some Basic Equations and Their Graphs in 3D

51. 3D Coordinate System
x
z+
The linear equations:
The graphs of ax + by + cz = d are
planes. Use the intercepts to graph:
Set x=y=0 to get the z intercept,
set x=z=0 to get the y intercept,
set y=z=0 to get the x intercept.
Example: Sketch 2x – y + 2z = 4
Set x=y=0  z=2, get (0, 0, 2)
Set x=z=0  y=–4, get (0, –4, 0)
Set y=z=0  x=2, get (2, 0, 0).
y
Three points determine a plane. Plot these three
intercepts and the plane containing them is the graph.
Some Basic Equations and Their Graphs in 3D
(2, 0, 0)
(0, –4, 0)
(0, 0, 2)

52. 3D Coordinate System
x
z+
The linear equations:
The graphs of ax + by + cz = d are
planes. Use the intercepts to graph:
Set x=y=0 to get the z intercept,
set x=z=0 to get the y intercept,
set y=z=0 to get the x intercept.
Example: Sketch 2x – y + 2z = 4
Set x=y=0  z=2, get (0, 0, 2)
Set x=z=0  y=–4, get (0, –4, 0)
Set y=z=0  x=2, get (2, 0, 0).
y
Three points determine a plane. Plot these three
intercepts and the plane containing them is the graph.
Some Basic Equations and Their Graphs in 3D
(2, 0, 0)
(0, –4, 0)
(0, 0, 2)

53. 3D Coordinate System
Some Basic Equations and Their Graphs in 3D
General Cylinders (equation with a missing variable)

54. 3D Coordinate System
Some Basic Equations and Their Graphs in 3D
General Cylinders (equation with a missing variable)
If an equation doesn't contain a particular
variable, that variable may take on any
value. To draw its graph, draw the 2D graph
in the coordinate plane of the variables in
the equation.
Example: Sketch z = x2

55. 3D Coordinate System
y
z
x
Some Basic Equations and Their Graphs in 3D
General Cylinders (equation with a missing variable)
If an equation doesn't contain a particular
variable, that variable may take on any
value. To draw its graph, draw the 2D graph
in the coordinate plane of the variables in
the equation.
Example: Sketch z = x2
Draw z = x2 in the xz–plane which is a parabola.

56. 3D Coordinate System
Some Basic Equations and Their Graphs in 3D
General Cylinders (equation with a missing variable)
If an equation doesn't contain a particular
variable, that variable may take on any
value. To draw its graph, draw the 2D graph
in the coordinate plane of the variables in
the equation.
Slide the 2D graph parallelwise
in the direction of the missing variable,
the surface it forms is the 3D graph.
Example: Sketch z = x2
Draw z = x2 in the xz–plane which is a parabola.
y
z
x

57. 3D Coordinate System
y
z
x
Some Basic Equations and Their Graphs in 3D
General Cylinders (equation with a missing variable)
If an equation doesn't contain a particular
variable, that variable may take on any
value. To draw its graph, draw the 2D graph
in the coordinate plane of the variables in
the equation.
Slide the 2D graph parallelwise
in the direction of the missing variable,
the surface it forms is the 3D graph.
Example: Sketch z = x2
Draw z = x2 in the xz–plane which is a parabola. Slide this
parabola in the y (the missing variable) direction we get the
(parabolic) cylinder–surface as shown.

58. 3D Coordinate System
y
z
x
Some Basic Equations and Their Graphs in 3D
General Cylinders (equation with a missing variable)
If an equation doesn't contain a particular
variable, that variable may take on any
value. To draw its graph, draw the 2D graph
in the coordinate plane of the variables in
the equation.
Slide the 2D graph parallelwise
in the direction of the missing variable,
the surface it forms is the 3D graph.
This is called a general cylinder.
Example: Sketch z = x2
Draw z = x2 in the xz–plane which is a parabola. Slide this
parabola in the y (the missing variable) direction we get the
(parabolic) cylinder–surface as shown.

59. 3D Coordinate System
Distance Formula in 3D:
The distance between (x1, y1, z1), (x2, y2, z2) is:
D =  Δx2 + Δy2 + Δz2
= (x2 – x1)2 + (y2 – y1)2 + (z2 – z1)2

60. 3D Coordinate System
Distance Formula in 3D:
The distance between (x1, y1, z1), (x2, y2, z2) is:
D =  Δx2 + Δy2 + Δz2
= (x2 – x1)2 + (y2 – y1)2 + (z2 – z1)2
Example: The distance between (2, –1, 1) and (1, –1, 3) is
Δx = 1, Δy = 0, Δz = –2, so D = 1+ 0 + 4 = 5

61. 3D Coordinate System
Distance Formula in 3D:
The distance between (x1, y1, z1), (x2, y2, z2) is:
D =  Δx2 + Δy2 + Δz2
= (x2 – x1)2 + (y2 – y1)2 + (z2 – z1)2
Example: The distance between (2, –1, 1) and (1, –1, 3) is
Δx = 1, Δy = 0, Δz = –2, so D = 1+ 0 + 4 = 5
Mid–Point Formula: The mid–point in 3D is computed
coordinate–wise so the mid–point of
(x1, y1, z1), (x2, y2, z2) is:
(
x1+ x2 y1+ y2 z1+ z2
2 2 2, , )

62. 3D Coordinate System
Distance Formula in 3D:
The distance between (x1, y1, z1), (x2, y2, z2) is:
D =  Δx2 + Δy2 + Δz2
= (x2 – x1)2 + (y2 – y1)2 + (z2 – z1)2
Example: The distance between (2, –1, 1) and (1, –1, 3) is
Δx = 1, Δy = 0, Δz = –2, so D = 1+ 0 + 4 = 5
Mid–Point Formula: The mid–point in 3D is computed
coordinate–wise so the mid–point of
(x1, y1, z1), (x2, y2, z2) is:
(
x1+ x2 y1+ y2 z1+ z2
2 2 2, , )
y
z+
xA(2, 0 , 0)
B(1, 3, 4)

63. 3D Coordinate System
Distance Formula in 3D:
The distance between (x1, y1, z1), (x2, y2, z2) is:
D =  Δx2 + Δy2 + Δz2
= (x2 – x1)2 + (y2 – y1)2 + (z2 – z1)2
Example: The distance between (2, –1, 1) and (1, –1, 3) is
Δx = 1, Δy = 0, Δz = –2, so D = 1+ 0 + 4 = 5
Mid–Point Formula: The mid–point in 3D is computed
coordinate–wise so the mid–point of
(x1, y1, z1), (x2, y2, z2) is:
(
x1+ x2 y1+ y2 z1+ z2
2 2 2, , )
y
z+
xA(2, 0 , 0)
B(1, 3, 4)
mid–pt
(3/2, 3/2, 2)
(3/2, 3/2, 0)

64. 3D Coordinate System
Equations of Spheres
Some Basic Equations and Their Graphs in 3D

65. 3D Coordinate System
Equations of Spheres
Some Basic Equations and Their Graphs in 3D
The equation of the sphere with radius r,
centered at (a, b ,c) is
(x – a)2 + (y – b)2 + (z – c)2 = r2.
In particular x2 + y2 + z2 = r2 is the
sphere centered at (0, 0, 0) with radius r.

66. 3D Coordinate System
x
z+
Equations of Spheres
y
Some Basic Equations and Their Graphs in 3D
The equation of the sphere with radius r,
centered at (a, b ,c) is
(x – a)2 + (y – b)2 + (z – c)2 = r2.
In particular x2 + y2 + z2 = r2 is the
sphere centered at (0, 0, 0) with radius r.
(a, b, c)
r
r
(x–a)2+(y–b)2+(z–c)2=r2

67. 3D Coordinate System
x
z+
Equations of Spheres
y
Some Basic Equations and Their Graphs in 3D
The equation of the sphere with radius r,
centered at (a, b ,c) is
(x – a)2 + (y – b)2 + (z – c)2 = r2.
In particular x2 + y2 + z2 = r2 is the
sphere centered at (0, 0, 0) with radius r.
(a, b, c)
r
r
(x–a)2+(y–b)2+(z–c)2=r2
x2+y2+z2=r2
Equations of Ellipsoid
The graph of the equation
is the ellipsoid centered at (a, b ,c),
with
x–radius=r, y–radius=s, z–radius=t
(x – a)2 (y – b)2 (z – c)2
r2 s2 t2 = 1++

68. 3D Coordinate System
x
z+
Equations of Spheres
y
Some Basic Equations and Their Graphs in 3D
The equation of the sphere with radius r,
centered at (a, b ,c) is
(x – a)2 + (y – b)2 + (z – c)2 = r2.
In particular x2 + y2 + z2 = r2 is the
sphere centered at (0, 0, 0) with radius r.
(a, b, c)
r
r
(x–a)2+(y–b)2+(z–c)2=r2
x2+y2+z2=r2
Equations of Ellipsoid
The graph of the equation
is the ellipsoid centered at (a, b ,c),
with
x–radius=r, y–radius=s, z–radius=t
(x – a)2 (y – b)2 (z – c)2
r2 s2 t2 = 1++
x
z+
y
(a, b, c)
r
s
t

69. 3D Coordinate System
Example: a. Find the equation of the sphere
which has (2, 1, 3), (4, 3, –5) as a diameter.
b. What is the highest point on this sphere?

70. 3D Coordinate System
Example: a. Find the equation of the sphere
which has (2, 1, 3), (4, 3, –5) as a diameter.
The center of the sphere is the mid–point of the
two given points. Use the mid–point formula,
we've the center = (3, 2, –1).
b. What is the highest point on this sphere?

71. 3D Coordinate System
Example: a. Find the equation of the sphere
which has (2, 1, 3), (4, 3, –5) as a diameter.
The center of the sphere is the mid–point of the
two given points. Use the mid–point formula,
we've the center = (3, 2, –1). The radius is half
of the length of the diameter, so using the
distance formula r = ½22+22+82 = 32 and the
equation is (x – 3)2 + (y – 2)2 + (z + 1)2 = 18.
b. What is the highest point on this sphere?

72. 3D Coordinate System
Example: a. Find the equation of the sphere
which has (2, 1, 3), (4, 3, –5) as a diameter.
The center of the sphere is the mid–point of the
two given points. Use the mid–point formula,
we've the center = (3, 2, –1). The radius is half
of the length of the diameter, so using the
distance formula r = ½22+22+82 = 32 and the
equation is (x – 3)2 + (y – 2)2 + (z + 1)2 = 18.
b. What is the highest point on this sphere?
The highest point is 32 above the center
so it’s (3, 2, –1+32).