find an implicit equation for a curve surface or hypersurface given by parametric equations - Maple Help

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algcurves[implicitize] - find an implicit equation for a curve surface or hypersurface given by parametric equations

Calling Sequence

implicitize( peqs, params, d, options )

Parameters

peqs

-

list of equations of the form name=formula

params

-

list of equations of the form name=range

d

-

total degree of implicit polynomial to be found

options

-

(optional) equation(s) of the form option=name where option is one of maxIters, symbolic, tol, or useFNV

Description

• 

The implicitize(peqs, params, d, options) function computes an implicit (polynomial) equation for the algebraic curve, surface, or hypersurface defined by the parametric equations peqs.

• 

The output is given as a set that can be empty or have one or more elements.

• 

The implicitize routine returns an implicit equation for the degree d if it exists.

• 

The output will be exact or approximate, depending on the optional parameters controlled by the user.

• 

The options argument can contain one or more of the following equations.

  

maxIters = positive_integer

  

The maxIters option controls the maximum number of iterations performed by findNullVector.

  

symbolic = true or false

  

If the symbolic option is set to true, all integrations are performed symbolically. If the symbolic option is set to false, all integrations are performed numerically. The default value is false.

  

tol = expression

  

The tol option controls the tolerance used in findNullVector. The default value is 10eDigits.

  

useFNV = true or false

  

If the useFNV option is set to true, the implicitize routine uses findNullVector. If the useFNV option is set to false, the implicitize routine uses NullSpace instead of findNullVector. The default value is true.

• 

The implicitize routine accepts parametric equations that:

  

[1]  are polynomials, rational functions, trigonometric functions, functions involving square roots, or functions in polar form;

  

[2]  have coefficients from Q, parameters, or algebraic numbers; and

  

[3]  define multi-parameter families of curves, surfaces of hypersurfaces.

• 

The implicitize routine is based on the algorithm described in the paper: Robert M. Corless, Mark W. Giesbrecht, Ilias S. Kotsireas, Stephen M. Watt. "Numerical implicitization of parametric hypersurfaces with linear algebra." AISC'2000 Proceedings, Madrid, Spain. LNAI 1930 (in print).

Examples

withalgcurves:

peqs:=x=cosθ,y=sinθ

peqs:=x=cosθ,y=sinθ

(1)

implicitizepeqs,θ=0..π,2,symbolic=true,useFNV=false

x2+y21

(2)

peqs:=x=1t2+1,y=tt2+1

peqs:=x=1t2+1,y=tt2+1

(3)

implicitizepeqs,t=0..1,2,symbolic=true,useFNV=false

x2+y21

(4)

Peqs:=x=3t2t3+1,y=3tt3+1

Peqs:=x=3t2t3+1,y=3tt3+1

(5)

Digits:=25

Digits:=25

(6)

ee:=implicitizePeqs,t=0..1,3,symbolic=true

ee:=7.17575513080306601056700010-201.84588317962365134869416110-18y1.62036822677304905453680310-16y20.3015113445777636810112419y3+5.20833992024645202272212110-16x+0.9045340337332909517727017yx+3.41797954428839319634818310-16xy26.57194964762560012622241110-16x23.33697930400243922377984610-16x2y0.3015113445777633127855855x3

(7)

eq:=fnormalee,4

eq:=0.3015y3+0.9045yx0.3015x3

(8)

sm:=minopmapxx→xx,coeffsexpandeq

sm:=0.3015

(9)

fnormalsimplifyeqsm,3

y3+3.00yxx3

(10)

simplifyeval,Peqs

0.

(11)

implicitizex=rt,y=rt2,z=r2,r=0..1,t=0..1,4,symbolic=true,useFNV=false

x4y2z

(12)

peqs:=x=αtanφ,y=βcosφ2

peqs:=x=αtanφ,y=βcosφ2

(13)

implicitizepeqs,φ=π3..π4,3,symbolic=true,useFNV=false

α2β+α2y+x2y

(14)

simplifyeval,peqs

0

(15)

peqs:=x=sin3θcosθ,y=sin3θsinθ

peqs:=x=sin3θcosθ,y=sin3θsinθ

(16)

implicitizepeqs,θ=0..π,4,symbolic=true,useFNV=false

x4+2x2y2+y43x2y+y3

(17)

peqs:=x=t+s+r,y=ts+sr+rt,z=tsr,w=t3+s3+r3

peqs:=x=t+s+r,y=rs+rt+st,z=tsr,w=r3+s3+t3

(18)

implicitizepeqs,t=0..1,s=0..1,r=0..1,3,symbolic=true,useFNV=false

x33xyw+3z

(19)

simplifyeval,peqs

0

(20)

peqs:=x=tβt21+t22,y=t2βt21+t22

peqs:=x=tt2+βt2+12,y=t2t2+βt2+12

(21)

implicitizepeqs,t=0..2,2,symbolic=true,useFNV=false

(22)

implicitizepeqsβ=3|peqsβ=3,t=0..2,4,symbolic=true,useFNV=false

x4+2x2y2+y43x2y+y3

(23)

implicitizepeqsβ=4|peqsβ=4,t=0..2,4,symbolic=true,useFNV=false

x4+2x2y2+y44x2y+y3

(24)

p:=x4βyx2+2x2y2+y3+y4

p:=βx2y+x4+2x2y2+y4+y3

(25)

simplifyevalp,peqs

0

(26)

peqs:=x=3t22ttk,y=3t2tk

peqs:=x=3t22ttk,y=3t2tk

(27)

implicitizepeqsk=4|peqsk=4,t=1..2,4,symbolic=true,useFNV=false

x43xy2+2y3

(28)

implicitizepeqsk=5|peqsk=5,t=1..2,5,symbolic=true,useFNV=false

x53xy3+2y4

(29)

implicitizepeqsk=6|peqsk=6,t=1..2,6,symbolic=true,useFNV=false

x63xy4+2y5

(30)

p:=3t2;q:=tk;xtk:=tpq;ytk:=pq

p:=3t2

q:=tk

xtk:=t3t2tk

ytk:=3t2tk

(31)

exk:=xk3xyk2+2yk1

exk:=xk3xyk2+2yk1

(32)

dd:=evalexk,x=xtk,y=ytk

dd:=t3t2tkk3t3t23t2tkk2tk+23t2tkk1

(33)

simplifydd

t1k3t2k3t1+k3t2tkk3t2+23t2tkktk3t2

(34)

aliasβ=b

β

(35)

aliasγ=c

β,γ

(36)

peqs:=x=β+cos2θ,y=γ+sin2θ

peqs:=x=b+cos2θ,y=c+sin2θ

(37)

implicitizepeqs,θ=0..π,2,symbolic=true,useFNV=false

b+c12yc+y22xb+x2

(38)

simplifyeval,peqs

0

(39)

See Also

abs, algcurves, alias, coeffs, convert, Digits, eval, expand, fnormal, map, min, op, simplify, sqrt, trig


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