General Tent

Mandelbrot & Julia

by Kerry Mitchell, 05 October 2000

Basic information

The standard tent map is defined for a real variable x, such that:
xnew = xold, if xold <= 1, and
xnew = 2 - xold, if xold > 1.
When xnew is plotted vs. xold, a tent shape results, with the peak at the critical point, xold = 1.

To use this formula with a complex variable z, a way is needed to generate a real value for the tent map.  These formulas offer several:  magnitude of z, real part, imaginary part, imaginary/real, and real*imaginary.  Also, different rotational modes are offered.  For example, instead of using just the real part of z, maybe the 45 degree line y = x is taken as the new "x" axis for the tent map.  This rotation can be done is several ways.

Suggested coloring formulas

There can be a lot of structure (or noise) created with these formulas, so stick with basic iteration outside coloring until you get the hang of things.  For inside lake areas, anything that works well with the standard Mandelbrot and Julia formulas should work fine.

Parameters

Hints

Sample images

general-tent-sample1 { ; copyright Kerry Mitchell 01oct2000
;
; Sample of the General Tent formula
; using 'imaginary part' r type and 'constant' rotation
; of 45 degrees
;
fractal:
  title="General Tent Sample 1" width=480 height=480 author="Kerry Mitchell"
  created="October 1, 2000" numlayers=2
layer:
  method=multipass caption="New Layer 2" opacity=100 visible=yes alpha=no
  mergemode=multiply
mapping:
  center=0/0 magn=2.4 angle=340
formula:
  filename="lkm.ufm" entry="general-tent-julia" maxiter=1000 percheck=off
  p_julparam=1.0513704938994191/0.498206243061781633 p_bailout=1000000
  p_rtype="imag part" p_rottype="constant" p_rotamount=45
  p_rotunit="degrees"
inside:
  transfer=none repeat=yes
outside:
  filename="lkm.ucl" entry="rose-range-lite" density=3 transfer=linear
  repeat=yes p_scale=0.75 p_rangewidth=0.5 p_colorby="last angle" p_ac=1
  p_bc=3 p_as=0 p_bs=0 p_curvecenter=0/0 p_rot=0
gradient:
  smooth=yes numnodes=4 index=0 color=4308801 index=100 color=0 index=200
  color=4308801 index=300 color=16777215
layer:
  method=multipass caption="New Layer 1" visible=yes alpha=no
mapping:
  center=0/0 magn=2.4 angle=340
formula:
  filename="lkm.ufm" entry="general-tent-julia" maxiter=1000 percheck=off
  p_julparam=1.0513704938994191/0.498206243061781633 p_bailout=1000000
  p_rtype="imag part" p_rottype="constant" p_rotamount=45
  p_rotunit="degrees"
inside:
  transfer=none repeat=yes
outside:
  filename="lkm.ucl" entry="rose-range-lite" transfer=linear repeat=yes
  p_scale=0.75 p_rangewidth=0.5 p_colorby="last magnitude" p_ac=1 p_bc=3
  p_as=0 p_bs=0 p_curvecenter=0/0 p_rot=0
gradient:
  smooth=yes numnodes=4 index=0 color=0 index=100 color=4308801 index=200
  color=16777215 index=300 color=4308801
}

general-tent-sample2 { ; copyright Kerry Mitchell 02oct2000
;
; Sample of General Tent formula
; using 'magnitude' r type and 'none' rotation
;
fractal:
  title="General Tent Sample 2" width=480 height=480 author="Kerry Mitchell"
  created="October 2, 2000" numlayers=1
layer:
  caption="Layer 1" visible=yes alpha=no
mapping:
  center=1.23/1.55 magn=0.7 angle=0
formula:
  filename="lkm.ufm" entry="general-tent-julia" maxiter=1000 percheck=off
  p_julparam=-0.43033100872625215/0.937762375451733166 p_bailout=1000000
  p_rtype="magnitude" p_rottype="none" p_rotamount=45 p_rotunit="degrees"
inside:
  transfer=none repeat=yes
outside:
  density=3 transfer=sqrt repeat=yes
gradient:
  smooth=yes numnodes=4 index=84 color=0 index=183 color=4292799 index=283
  color=16777215 index=383 color=4292799
}

general-tent-sample3 { ; copyright Kerry Mitchell 02oct2000
;
; Sample of General Tent formula
; using 'real part' r type and 'progressive' rotation type
; of 180 degrees
;
fractal:
  title="General Tent Sample 3" width=600 height=200 author="Kerry Mitchell"
  created="October 2, 2000" numlayers=2
layer:
  method=multipass caption="New Layer 2" opacity=100 visible=yes alpha=no
  mergemode=difference
mapping:
  center=1/0 magn=1 angle=315
formula:
  filename="lkm.ufm" entry="general-tent-julia" maxiter=1000 percheck=off
  p_julparam=0/1.41421356237309505 p_bailout=1000000 p_rtype="real part"
  p_rottype="progressive" p_rotamount=180 p_rotunit="degrees"
inside:
  transfer=none repeat=yes
outside:
  filename="lkm.ucl" entry="conic-lite" transfer=sqrt repeat=yes
  p_type="circle" p_center=0/0 p_theta=45 p_radius=1 p_xaxis=1.5
  p_yaxis=0.75 p_colorby="minimum distance"
gradient:
  smooth=yes numnodes=4 index=50 color=12542017 index=150 color=16777215
  index=250 color=4300991 index=349 color=0
layer:
  method=multipass caption="New Layer 1" visible=yes alpha=no
mapping:
  center=1/0 magn=1 angle=315
formula:
  filename="lkm.ufm" entry="general-tent-julia" maxiter=1000 percheck=off
  p_julparam=0/1.41421356237309505 p_bailout=1000000 p_rtype="real part"
  p_rottype="progressive" p_rotamount=180 p_rotunit="degrees"
inside:
  transfer=none repeat=yes
outside:
  filename="lkm.ucl" entry="conic-lite" transfer=sqrt repeat=yes
  p_type="line" p_center=1/0 p_theta=-45 p_radius=1 p_xaxis=1.5
  p_yaxis=0.75 p_colorby="minimum distance"
gradient:
  smooth=yes numnodes=2 index=50 color=16777215 index=250 color=0
}