Axiom Projective Space Geometry Example

The example on this page attempts to show, not just how to create an image in FriCAS but also how to use it with other programs.

I wanted to make the image on the right to illustrate the topology of projective space discussed on this page.

We can use FriCAS to generate a parametric surface.

In this case, for all x,y then z=x*y

We can do this by using the following SPAD code:

This code generates the 3D surface and also a 2D grid which we can use to texture the surface.

)expose SCartesian
)expose SArgand
)expose SConformal
)expose SceneIFS
)expose SceneNamedPoints
)expose STransform
)expose SBoundary
)expose ExportXml

DF ==> DoubleFloat
PT ==> SCartesian(3)
xyfn(x:DF,y:DF):DF == x*y
view := boxBoundary(sipnt(0,-500)$PT,sipnt(1200,500)$PT)
sc := createSceneRoot(view)$Scene(PT)
tr2 := addSceneTransform(sc,identity()$STransform(PT))$Scene(PT)
gd := addPlot2Din3D(tr2,xyfn,-1..1,-1..1,49)$Scene(PT)
writeX3d(sc,"projectiveSpace.x3d")


PT ==> SArgand
view2 := boxBoundary(sipnt(0,-500)$PT,sipnt(1200,500)$PT)
sc := createSceneRoot(view2)$Scene(PT)
gd := addSceneGrid(sc,view2)$Scene(PT)
writeSvg(sc,"grid.svg")

First we use a program called Inkscape to convert the grid from a SVG file to a PNG file. (that is convert from a vector format to a raster format)

In Inkscape

Start Inkscape

open grid.svg

export it as grid.png

We now import this into Blender and do the following:

In Blender

Start blender and remove default cube
select camera view
import projectiveSpace.x3d file
rotate shape so we can see the corners starting to move together.

Now select the object and add material to it
remove spectial effects
select material and add texture
create new image and animation
open (not new) file: grid.png
in Render tab Set alpha channel

Render