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geom3d

  

sphere

  

define a sphere

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

sphere(s, [A, B, C, D], n, 'centername'=m)

sphere(s, [A, B], n, 'centername'=m)

sphere(s, [A, rad], n, 'centername'=m)

sphere(s, [A, p], n, 'centername'=m)

sphere(s, eqn, n, 'centername'=m)

Parameters

s

-

the name of the sphere

A, B, C, D

-

points

p

-

plane

rad

-

number which is the radius of the sphere

eqn

-

the algebraic representation of the sphere (i.e., a polynomial or an equation)

n

-

(optional) list of three names representing the names of the axes

'centername'=m

-

(optional) m is a name of the center of the sphere to be created

Description

• 

A sphere is the set of all points in space that have the same distance from the center.

• 

A sphere s can be defined as follows:

– 

from four points A, B, C, and D.

– 

from the two points which make a diameter of the sphere s.

– 

from the center of s and its radius. The input is a list of two elements where the first element is a point, the second element is a number.

– 

from a point A and a plane p. The sphere defined is the one with center A and is tangent to the plane p.

– 

from its algebraic representation eqn. The input is an equation or a polynomial. In case the optional argument n is not given, if the environment variables _EnvXName, _EnvYName and _EnvZName are assigned to three names, these three names will be used as the names of the axes. If not, Maple will prompt for input of the names of the axes.

• 

To access the information relating to a sphere s, use the following function calls:

form(s)

returns the form of the geometric object (i.e., sphere3d if s is a sphere).

center(s)

returns the name of the center of s.

radius(s)

returns the radius of s.

area(s)

returns the surface area of s.

volume(s)

returns the volume of s.

Equation(s)

returns the equation that represents the sphere.

xname(s), yname(s), or zname(s)

returns the name of the axis; or FAIL if the axis is not assigned to any name.

detail(s)

returns a detailed description of the given sphere s.

• 

The command with(geom3d,sphere) allows the use of the abbreviated form of this command.

Examples

withgeom3d:

_EnvXNamex:_EnvYNamey:_EnvZNamez:

Find the equation of the sphere through the origin o and three points A,B,C whose coordinates are a,0,0,0,b,0,0,0,c.

Make sure that A,B,C are not the same as the origin

assumea,real,a0,b,real,b0,c,real,c0:

pointo,0,0,0,pointA,a,0,0,pointB,0,b,0,pointC,0,0,c:

spheres,o,A,B,C

s

(1)

Equations

a~xb~yc~z+x2+y2+z2=0

(2)

a'a':b'b':c'c':

C is a fixed point on OZ and U,V are variable points on OX,OY respectively. Find the locus of a point P when the lines PU,PV,PC are mutually at right angles.

Make necessary assumptions:

assumet1,real,t2,real,t3,real,t10,t20,t30,x0,y0,z0:

Define points C,U,V

pointC,0,0,t1,pointU,t2,0,0,pointV,0,t3,0:

Let P=x,y,z

pointP,x,y,z:

Define lines PU,PV, and PC

linePU,P,U,linePV,P,V,linePC,P,C:

It is obvious that there is not enough information to determine if PU,PV, and PC are mutually at right angles. Check for the conditions that make them mutually at right angles:

ArePerpendicularPU,PV,'cond1';ArePerpendicularPU,PC,'cond2'

FAIL

FAIL

(3)

ArePerpendicularPV,PC,'cond3'

FAIL

(4)

Take a look at the conditions:

cond1,cond2,cond3

x~+t2~x~y~t3~y~+z~2=0,x~+t2~x~+y~2z~t1~z~=0,x~2y~t3~y~z~t1~z~=0

(5)

expandcond2expandcond1

t1~z~+t3~y~=0

(6)

the above shows that yt3=zt1. Substitute this identity into cond3

subsyt3=zt1,lhsexpandcond3

2t1~z~+x~2+y~2+z~2

(7)

define a sphere s from the above condition:

spheres,,x,y,z

s

(8)

check:

coordinatescenters=coordinatesC

0,0,t1~=0,0,t1~

(9)

radiuss=distancecenters,pointo,0,0,0

t1~2=t1~2

(10)

See Also

geom3d[ArePerpendicular]

geom3d[center]

geom3d[intersection]

geom3d[inversion]

geom3d[polar]

geom3d[pole]

geom3d[radius]