Axiom - Geometry Framework Examples

Before we can use the scenegraph we need to make the domains available.

This is very tedious, it would be better if this was done by default.

(1) -> )expose SCartesian

   SCartesian is now explicitly exposed in frame frame1
(1) -> )expose SArgand

   SArgand is now explicitly exposed in frame frame1
(1) -> )expose SConformal

   SConformal is now explicitly exposed in frame frame1 
(1) -> )expose SceneIFS

   SceneIFS is now explicitly exposed in frame frame1 
(1) -> )expose SceneNamedPoints

   SceneNamedPoints is now explicitly exposed in frame frame1 
(1) -> )expose STransform

   STransform is now explicitly exposed in frame frame1 
(1) -> )expose SBoundary

   SBoundary is now explicitly exposed in frame frame1 
(1) -> )expose ExportXml

   ExportXml is now explicitly exposed in frame frame1 

Example 1 - Two dimensional plots with scale.

In this example we have 2 scalar functions superimposed (sine and tangent) plotted in two dimensions. The tangent function has been clipped to values between -1 and +1. The coordinate system is SCartesian(2) which represents two dimensional cartesian coordinates. We also have a grid, numerical axes and text annotation.

DF ==> DoubleFloat
PT ==> SCartesian(2)
fnsin(x:DF):DF == sin(x)
fntan(x:DF):DF == tan(x)
view := boxBoundary(sipnt(-1,-1)$PT,sipnt(3,1)$PT)
sc := createSceneRoot(view)$Scene(PT)
gd := addSceneGrid(sc,view)$Scene(PT)
addSceneRuler(sc,"HORIZONTAL"::Symbol,spnt(0::DF,-0.1::DF)$PT,view)$Scene(PT)
addSceneRuler(sc,"VERTICAL"::Symbol,spnt(-0.1::DF,0::DF)$PT,view)$Scene(PT)
mt1 := addSceneMaterial(sc,3::DF,"blue","green")$Scene(PT)
ln1 := addPlot1Din2D(mt1,fnsin,0..3::DF,49)$Scene(PT)
mt2 := addSceneMaterial(sc,3::DF,"green","green")$Scene(PT)
bb := boxBoundary(sipnt(0,-1)$PT,sipnt(3,1)$PT)
bb2 := addSceneClip(mt2,bb)$Scene(PT)
ln2 := addPlot1Din2D(bb2,fntan,0..3::DF,49)$Scene(PT)
addSceneText(sc,"sin(theta)",32::NNI,_
                   spnt(0.5::DF,0.4::DF)$PT)$Scene(PT)
addSceneText(sc,"tan(theta)",32::NNI,_
                   spnt(0.1::DF,0.6::DF)$PT)$Scene(PT)
writeSvg(sc,"example1.svg")

axiom plot example 1

Example 2 - Plotting using complex numbers.

Here the coordinate system is SArgand which represents two dimensions on an argand plane. As we can see the sine function and background grid can be drawn in exactly the same way as with cartesian coordinates. But an exponential function is applied to the tangent function.

DF ==> DoubleFloat
PT ==> SArgand
fnsin(x:DF):DF == sin(x/100::DF)*400::DF
fntan(x:DF):DF == tan(x/100::DF)*400::DF
C ==> Complex DoubleFloat
fnexp(x:C):C == exp(x::C/complex(100::DF,0::DF))
view := boxBoundary(sipnt(0,-500)$PT,sipnt(1200,500)$PT)
sc := createSceneRoot(view)$Scene(PT)
gd := addSceneGrid(sc,view)$Scene(PT)
mt1 := addSceneMaterial(sc,3::DF,"blue","green")$Scene(PT)
ln1 := addPlot1Din2D(mt1,fnsin,0..1000::DF,49)$Scene(PT)
mt2 := addSceneMaterial(sc,3::DF,"green","green")$Scene(PT)
tr2 := addSceneTransform(mt2,stransform(fnexp::(C -> C))_
       )$Scene(PT)
bb := boxBoundary(sipnt(100,-400)$PT,sipnt(1100,400)$PT)
bb2 := addSceneClip(tr2,bb)$Scene(PT)
ln2 := addPlot1Din2D(bb2,fntan,0..1000::DF,49)$Scene(PT)
tx := addSceneText(sc,"sin(theta)",32::NNI,_
       sipnt(200,400)$PT)$Scene(PT)
writeSvg(sc,"example2.svg")

svg plot

Example 3 - Function of a Complex Number.

Here we use 'addScenePattern1' which consists of horizontal (red) and vertical (blue) lines. We put this under a transform node containing an exponential function which produces the pattern shown here.

DF ==> DoubleFloat
PT ==> SArgand
C ==> Complex DoubleFloat
fnexp(x:C):C == exp(x::C/complex(100::DF,0::DF))
view := boxBoundary(sipnt(0,-500)$PT,sipnt(1200,500)$PT)
sc := createSceneRoot(view)$Scene(PT)
tr2 := addSceneTransform(sc,stransform(fnexp::(C -> C))_
       )$Scene(PT)
gd := addScenePattern(tr2,"GRID"::Symbol,20,view)$Scene(PT)
writeSvg(sc,"example3.svg")

svg plot

This vertical and horizontal line pattern is useful to test transforms as it indicates whether angles are preserved (lines still meet at right angles) however we can use other test patterns, for instance, this Sierpinski fractel:

DF ==> DoubleFloat
PT ==> SArgand
C ==> Complex DoubleFloat
fnexp(x:C):C == exp(x::C/complex(100::DF,0::DF))
view := boxBoundary(sipnt(0,-500)$PT,sipnt(1200,500)$PT)
sc := createSceneRoot(view)$Scene(PT)
tr2 := addSceneTransform(sc,stransform(fnexp::(C -> C))_
       )$Scene(PT)
gd := addScenePattern(tr2,"SIERPINSKI"::Symbol,4,view)$Scene(PT)
writeSvg(sc,"example32.svg")

svg plot

Example 4 - Parametric Two dimensional plot.

Here we use the parametric function sin(t),cos(t) to draw part of a circle.

DF ==> DoubleFloat
PT ==> SCartesian(2)
PPC  ==> ParametricPlaneCurve(DF -> DF)
xsin(t:DF):DF == sin(t)*500::DF
xcos(t:DF):DF == cos(t)*500::DF + 500::DF
view := boxBoundary(sipnt(0,-500)$PT,sipnt(1200,500)$PT)
sc := createSceneRoot(view)$Scene(PT)
tr2 := addSceneTransform(sc,_
       identity()$STransform(PT))$Scene(PT)
gd := addPlot1Din2Dparametric(tr2,curve(xsin,_
                    xcos)$PPC,0..6::DF,49)$Scene(PT)
writeSvg(sc,"example4.svg")

svg plot

Example 5 - Three Dimensional Box.

Here we have a three dimensional cube which is rotated.

DF ==> DoubleFloat
PT ==> SCartesian(3)
view := boxBoundary(sipnt(0,-500)$PT,sipnt(1200,500)$PT)
sc := createSceneRoot(view)$Scene(PT)
tr2 := addSceneTransform(_
       sc,stransform([[0.7::DF,0.7::DF,0::DF,0::DF],_
                    [-0.7::DF,0.7::DF,0::DF,0::DF],_
                    [0::DF,0::DF,1::DF,0::DF],_
                    [0::DF,0::DF,0::DF,1::DF]]_
                      )$STransform(PT))$Scene(PT)
box := addSceneBox(tr2,1::DF)$Scene(PT)
writeX3d(sc,"example5.x3d")

svg plot

Example 6 - Parametric Three Dimensional plot.

A parametric function, this time in three dimensions.

DF ==> DoubleFloat
PT ==> SCartesian(3)
PSC  ==> ParametricSpaceCurve(DF -> DF)
ysin(t:DF):DF == sin(t)*3::DF
ycos(t:DF):DF == cos(t)*3::DF
ytan(t:DF):DF == t::DF
view := boxBoundary(sipnt(-500,-500)$PT,sipnt(500,500)$PT)
sc := createSceneRoot(view)$Scene(PT)
tr2 := addSceneTransform(sc,_
       identity()$STransform(PT))$Scene(PT)
gd := addPlot1Din3Dparametric(tr2,curve(ysin,_
               ycos,ytan)$PSC,0..(6::DF),49)$Scene(PT)
writeX3d(sc,"example6.x3d")

svg plot

Example 7 - Three Dimensional Surface.

A three dimensional surface.

DF ==> DoubleFloat
PT ==> SCartesian(3)
xyfn(x:DF,y:DF):DF == x*x-y*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,"example7.x3d")

svg plot

Further Information

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


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