elementary forms

formeless forms ?

The basic pFormes are in fact "formeless" forms, forms whose the control points are put anywhere in the space. To be more fair, these forms are not absolutely "formeless", the control points distribute their points in such a big number as we can wish it in positions completely determined by recursive application of simple linear combinations. These are « recursive multilinear forms » leading by reduction / diagonalisation to forms as complex as we wish it but their equations are always finished polynomes. The feature of this polynomes is that their coefficients are those of the Pascal's triangle, that's why they're named "pascalian forms". If you forgot what is the Pascal's triangle, consider this series of numbers:

                   1             a point     (pP1)
                   1 1           a segment   (pL2)
                   1 2 1         a parabola (pL3)
                   1 3 3 1       a cubic  (pL4)
                   1 4 6 4 1     ??? :) (pL5)
                   . . . . . . . . . . . .

You see? If you have mind more "physicist / mechanic" than mathematician, you can think about barycentric combinations of masses, or about the gravity centre of a set of massic points. All points of these forms are distributed in a kind of very natural equilibrium of masses. Addicts of statistics and probability can also think about the laws of distribution. These forms are "natural", they correspond to numerous natural phenomena translating the equilibrium of actions, masses, space occupation, optimization. Therefore, they are very general (see the page conics for more precision on this point).

an example

Here the code generating the first image :

def exemple()
	# a table of bidimensional points :
	surf = [[ [-0.5, 0.5, 0.0], [0.5, 0.5,-0.5] ], 
			[ [-0.5,-0.5,-0.5], [0.5,-0.5, 0.0] ] ]
	pS22_0 = PF.new( surf )	                              # a first pSurface
        pS22_0.stretch( [0.25,0.25], [0.75,0.75] )            # stretch
	pS22_1 = PF.new( surf )                               # a second pSurface
        pS22_1.translate( [0.0,0.0,0.5] )                     # moved
	pS22_1.rotate( [0,0,-15] )                            # rotate
	pV222 = PF.new( [ pS22_0.poles(), pS22_1.poles() ] )  # a pVolume
		pV222.build( 2 )                              # built
		pV222.draw( "V" )                             # displayed like a volume (6 faces)
	pS = pV222.diag()                                     # its diagonal pSurface 
	pS.stretch( [0.25,0.25], [0.75,0.75] )                # stretch
		pS.build( 3 )                                 # built
		pS.draw( "S" )                                # displayed with surfaces
	pL = pS.diag()                                        # its diagonal pCourbe 
	pL.stretch( [-1.0], [2.0] )                           # stretch
		pL.build( 5 )                                 # built
		pL.draw( "P" )                                # displayed 
end

explanation

A table of 4 points is built, surf, from which two identical curved facets are constructed (more known under the name of Hypars). The first facet is lightly reduced in its initial surface, it is not a homothetic transformation, the second is translated on the axe OZ and rotated of 15 ° on this axe. These two facets are combined in a curved cube pV222 which is shown (the 6 curved facets enveloping).The diagonal of this volume is a surface which is reduced (in its initial surface) and shown. The diagonal of this surface is a curve, stretched (in its initial curve) and displayed in form of series of points.

various things

A pCourbe, a pSurface, a pVolume are all linear barycentric combinations of points; they are created and treated similarly generating numerous properties. Ones can define hyper_pVolumes and beyond pFormes of any dimensions. What can be interest to work with such pFormes?

A pForme can be diagonalised, generating a pForme of lower dimension but more complex; for example the second diagonal of a cube (multilinear pForme) is a cubic (defined by a cubic polygon).
Conversely, a complex curve (built on a polygon of high degree) can be considered to be the umpteenth diagonal of a simpler pForme (one multilinear hyper_pVolume). This way of doing is common in mathematics and often rich in results.

PFormes can be linearly combined to produce new pFormes: for instance the combination pS1 + pS2 - pS3 produces a pSurface; and this pSurface is a surface of Coons if the limit curves of 3 pSurfaces are two-two convergent. So, we have a simple method to create a surface interpolating an border constituted by 4 curves.

This unicity also allows to treat global deformations of a complex pForme by embedding it in a pForme of higher dimension and lower complexity: for instance we can immerse a pSurface in a pCube constructed on 8 points (or more if necessary) and deform the pSurface on the whole by acting on the 8 points.

PFormes try to fill up a "lacuna" existing between simple forms as circle, conics, sphere, tore, and forms taken in all their generality (we can say formeless) studied in the lessons of differential geometry in universities. In classical approach, we study tangent plan in a point of a surface, then osculating quadric, and then it is big space...For those who heared about Taylor's serial developments (or of MacLaurin), they can say that pSurfaces pS22, pS32, pS33, pS34, etc are the first terms of a serial development of any form. And it is easier to manipulate one pS33 than an osculating quadric...

...on process

articles

A simplified approach on pForms is proposed in an article (in french !) published on january 2003 in the review "TANGENTE". You can download it in pdf (1,2Mo) : tangente. Another article is presented on the page straightaway, focusing on the concept of straight lines in curved forms.