From 0f53a7581030ae74c15782be1bbd863338cecd42 Mon Sep 17 00:00:00 2001 From: Simon Rohou Date: Tue, 10 Feb 2026 14:03:05 +0100 Subject: [PATCH 1/3] [doc] updated CtcInverse manual page for contraction on boxes --- .../contractors/analytic/ctcinverse.rst | 83 ++++++++++++------ .../contractors/analytic/himmelblau_boxes.png | Bin 0 -> 39850 bytes .../manual/contractors/analytic/src.cpp | 31 +++++-- doc/manual/manual/contractors/analytic/src.m | 30 +++++-- doc/manual/manual/contractors/analytic/src.py | 62 +++++++++++-- 5 files changed, 157 insertions(+), 49 deletions(-) create mode 100644 doc/manual/manual/contractors/analytic/himmelblau_boxes.png diff --git a/doc/manual/manual/contractors/analytic/ctcinverse.rst b/doc/manual/manual/contractors/analytic/ctcinverse.rst index c0f2959f9..545797153 100644 --- a/doc/manual/manual/contractors/analytic/ctcinverse.rst +++ b/doc/manual/manual/contractors/analytic/ctcinverse.rst @@ -92,7 +92,9 @@ To represent the vectors :math:`\mathbf{x}\in\mathbb{R}^n` consistent with the c :end-before: [ctcinv-1-end] :dedent: 0 -The contractor can be used as an operator to contract a 2d box :math:`[\mathbf{x}]`. It can also be involved in a paver in order to reveal the constraint: +The contractor can be used as an operator to contract a :math:`n`-d box :math:`[\mathbf{x}]`. + +Note that the contraction may be fast but not minimal, depending on your analytic expression. Therefore, you can also combine a ``CtcInverse`` with a ``CtcFixpoint`` to apply the contraction procedure repeatedly until a fixpoint is reached on the same box. The following code corresponds to the next figure, which shows contraction cases in blue and their fixpoint counterparts in red. .. tabs:: @@ -120,14 +122,10 @@ The contractor can be used as an operator to contract a 2d box :math:`[\mathbf{x :end-before: [ctcinv-2-end] :dedent: 0 -Which produces the following output: - -.. figure:: ./himmelblau_50.png +.. figure:: ./himmelblau_boxes.png :width: 400px - Outer approximation of the solution set for :math:`f(\mathbf{x})\in[50,50]`. This paving result reveals three connected subsets approximated with outer boxes. Blue parts are guaranteed not to contain solutions. - -We recall that for thick solution sets, one should prefer the use of the ``SepInverse`` separator in order to get both outer and inner approximations. For instance, the solution set associated with :math:`f(\mathbf{x})\leqslant 50` corresponds to: +As any other contractor, a ``CtcInverse`` can also be involved in a paver in order to reveal the constraint set: .. tabs:: @@ -155,6 +153,41 @@ We recall that for thick solution sets, one should prefer the use of the ``SepIn :end-before: [ctcinv-3-end] :dedent: 0 +Which produces the following output: + +.. figure:: ./himmelblau_50.png + :width: 400px + + Outer approximation of the solution set for :math:`f(\mathbf{x})\in[50,50]`. This paving result reveals three connected subsets approximated with outer boxes. Blue parts are guaranteed not to contain solutions. + +We recall that for thick solution sets, one should prefer the use of the ``SepInverse`` separator in order to get both outer and inner approximations. For instance, the solution set associated with :math:`f(\mathbf{x})\leqslant 50` corresponds to: + +.. tabs:: + + .. group-tab:: Python + + .. literalinclude:: src.py + :language: py + :start-after: [ctcinv-4-beg] + :end-before: [ctcinv-4-end] + :dedent: 4 + + .. group-tab:: C++ + + .. literalinclude:: src.cpp + :language: c++ + :start-after: [ctcinv-4-beg] + :end-before: [ctcinv-4-end] + :dedent: 4 + + .. group-tab:: Matlab + + .. literalinclude:: src.m + :language: matlab + :start-after: [ctcinv-4-beg] + :end-before: [ctcinv-4-end] + :dedent: 0 + .. figure:: ./himmelblau_50_inner.png :width: 400px @@ -176,24 +209,24 @@ can be easily approximated by the following union of contractors: .. literalinclude:: src.py :language: py - :start-after: [ctcinv-4-beg] - :end-before: [ctcinv-4-end] + :start-after: [ctcinv-5-beg] + :end-before: [ctcinv-5-end] :dedent: 4 .. group-tab:: C++ .. literalinclude:: src.cpp :language: c++ - :start-after: [ctcinv-4-beg] - :end-before: [ctcinv-4-end] + :start-after: [ctcinv-5-beg] + :end-before: [ctcinv-5-end] :dedent: 4 .. group-tab:: Matlab .. literalinclude:: src.m :language: matlab - :start-after: [ctcinv-4-beg] - :end-before: [ctcinv-4-end] + :start-after: [ctcinv-5-beg] + :end-before: [ctcinv-5-end] :dedent: 0 .. figure:: ./himmelblau_50_150_250.png @@ -323,24 +356,24 @@ When the constraint is a complement constraint :math:`\mathbf{f}(\mathbf{x})\not .. literalinclude:: src.py :language: py - :start-after: [ctcinv-5-beg] - :end-before: [ctcinv-5-end] + :start-after: [ctcinv-6-beg] + :end-before: [ctcinv-6-end] :dedent: 4 .. group-tab:: C++ .. literalinclude:: src.cpp :language: c++ - :start-after: [ctcinv-5-beg] - :end-before: [ctcinv-5-end] + :start-after: [ctcinv-6-beg] + :end-before: [ctcinv-6-end] :dedent: 4 .. group-tab:: Matlab .. literalinclude:: src.m :language: matlab - :start-after: [ctcinv-5-beg] - :end-before: [ctcinv-5-end] + :start-after: [ctcinv-6-beg] + :end-before: [ctcinv-6-end] :dedent: 0 @@ -358,24 +391,24 @@ The underlying analytic function can be accessed through ``.fnc()`` (useful for .. literalinclude:: src.py :language: py - :start-after: [ctcinv-6-beg] - :end-before: [ctcinv-6-end] + :start-after: [ctcinv-7-beg] + :end-before: [ctcinv-7-end] :dedent: 4 .. group-tab:: C++ .. literalinclude:: src.cpp :language: c++ - :start-after: [ctcinv-6-beg] - :end-before: [ctcinv-6-end] + :start-after: [ctcinv-7-beg] + :end-before: [ctcinv-7-end] :dedent: 4 .. group-tab:: Matlab .. literalinclude:: src.m :language: matlab - :start-after: [ctcinv-6-beg] - :end-before: [ctcinv-6-end] + :start-after: [ctcinv-7-beg] + :end-before: [ctcinv-7-end] :dedent: 0 Centered form option diff --git a/doc/manual/manual/contractors/analytic/himmelblau_boxes.png b/doc/manual/manual/contractors/analytic/himmelblau_boxes.png new file mode 100644 index 0000000000000000000000000000000000000000..bc6a70f0a9e564ce3289d7999fc26caa09dff850 GIT binary patch literal 39850 zcmeFZ^HYM5L4kVT*!-AV?^Z 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a/doc/manual/manual/contractors/analytic/src.cpp b/doc/manual/manual/contractors/analytic/src.cpp index 64a8f52ab..5e39b87c1 100644 --- a/doc/manual/manual/contractors/analytic/src.cpp +++ b/doc/manual/manual/contractors/analytic/src.cpp @@ -9,6 +9,7 @@ #include #include +#include #include #include #include @@ -29,22 +30,36 @@ TEST_CASE("CtcInverse - manual") // [ctcinv-1-end] // [ctcinv-2-beg] - DefaultFigure::pave({{-6,6},{-6,6}}, c, 1e-2); + IntervalVector z({{0,3.5},{0,1}}); + DefaultFigure::draw_box(z, {Color::blue(),Color::blue(.1)}); // prior to contraction + c.contract(z); + DefaultFigure::draw_box(z, Color::blue()); // after one CtcInverse contraction + // z == [ [1.84, 3.5] ; [0, 1] ] + + // Combining CtcInverse with a CtcFixpoint: + CtcFixpoint cfix(c); + c.contract(z); + DefaultFigure::draw_box(z, Color::red()); // after a fixed point contraction + // z == [ [1.84, 2.483] ; [0, 1] ] // [ctcinv-2-end] // [ctcinv-3-beg] - SepInverse s(f, {0,50}); - DefaultFigure::pave({{-6,6},{-6,6}}, s, 1e-2); + DefaultFigure::pave({{-6,6},{-6,6}}, c, 1e-2); // [ctcinv-3-end] // [ctcinv-4-beg] + SepInverse s(f, {0,50}); + DefaultFigure::pave({{-6,6},{-6,6}}, s, 1e-2); + // [ctcinv-4-end] + + // [ctcinv-5-beg] auto cu = CtcInverse(f,50) | CtcInverse(f,150) | CtcInverse(f,250); DefaultFigure::pave({{-6,6},{-6,6}}, cu, 1e-2); - // [ctcinv-4-end] + // [ctcinv-5-end] } { - // [ctcinv-5-beg] + // [ctcinv-6-beg] VectorVar x(2); AnalyticFunction f({x}, x[0]); @@ -54,19 +69,19 @@ TEST_CASE("CtcInverse - manual") IntervalVector y({{0.5,3},{-1,1}}); c.contract(y); // {{1,3},{-1,1}} // Only the first component is constrained by the not-in condition - // [ctcinv-5-end] + // [ctcinv-6-end] CHECK(y == IntervalVector({{1,3},{-1,1}})); } { - // [ctcinv-6-beg] + // [ctcinv-7-beg] VectorVar x(2); AnalyticFunction f({x}, x[0]-x[1]); CtcInverse c(f, 0); // c.fnc().input_size() == 2 // c.fnc().output_size() == 1 - // [ctcinv-6-end] + // [ctcinv-7-end] CHECK(c.fnc().input_size() == 2); CHECK(c.fnc().output_size() == 1); diff --git a/doc/manual/manual/contractors/analytic/src.m b/doc/manual/manual/contractors/analytic/src.m index 5461a6872..de2edd175 100644 --- a/doc/manual/manual/contractors/analytic/src.m +++ b/doc/manual/manual/contractors/analytic/src.m @@ -17,20 +17,34 @@ % [ctcinv-1-end] % [ctcinv-2-beg] -DefaultFigure().pave(IntervalVector({{-6,6},{-6,6}}), c, 1e-2); +z = IntervalVector({{0.0,3.5},{0.0,1.0}}); +DefaultFigure().draw_box(z, {Color().blue(),Color().blue(.1)}); % prior to contraction +z = c.contract(z); +DefaultFigure().draw_box(z, Color().blue()); % after one CtcInverse contraction +% z == [ [1.84, 3.5] ; [0, 1] ] + +% Combining CtcInverse with a CtcFixpoint: +cfix = CtcFixpoint(c); +z = c.contract(z); +DefaultFigure().draw_box(z, Color().red()); % after a fixed point contraction +% z == [ [1.84, 2.483] ; [0, 1] ] % [ctcinv-2-end] % [ctcinv-3-beg] -s = SepInverse(f, Interval(0,50)); -DefaultFigure().pave(IntervalVector({{-6,6},{-6,6}}), s, 1e-2); +DefaultFigure().pave(IntervalVector({{-6,6},{-6,6}}), c, 1e-2); % [ctcinv-3-end] % [ctcinv-4-beg] -cu = CtcUnion(CtcUnion(CtcInverse(f,50), CtcInverse(f,150)), CtcInverse(f,250)); -DefaultFigure().pave(IntervalVector({{-6,6},{-6,6}}), cu, 1e-2) +s = SepInverse(f, Interval(0,50)); +DefaultFigure().pave(IntervalVector({{-6,6},{-6,6}}), s, 1e-2); % [ctcinv-4-end] % [ctcinv-5-beg] +cu = CtcUnion(CtcUnion(CtcInverse(f,50), CtcInverse(f,150)), CtcInverse(f,250)); +DefaultFigure().pave(IntervalVector({{-6,6},{-6,6}}), cu, 1e-2) +% [ctcinv-5-end] + +% [ctcinv-6-beg] x = VectorVar(2); f = AnalyticFunction({x}, x(1)); @@ -40,14 +54,14 @@ y = IntervalVector({{0.5,3},{-1,1}}); c.contract(y); % [[1,3],[-1,1]] % Only the first component is constrained by the not-in condition -% [ctcinv-5-end] +% [ctcinv-6-end] assert(y==IntervalVector({{1,3},{-1,1}})); -% [ctcinv-6-beg] +% [ctcinv-7-beg] x = VectorVar(2); f = AnalyticFunction({x}, x(1)-x(2)); c = CtcInverse(f, 0); assert(c.fnc().input_size() == 2); assert(c.fnc().output_size() == 1); -% [ctcinv-6-end] \ No newline at end of file +% [ctcinv-7-end] \ No newline at end of file diff --git a/doc/manual/manual/contractors/analytic/src.py b/doc/manual/manual/contractors/analytic/src.py index a4cf162f3..76d9f8066 100644 --- a/doc/manual/manual/contractors/analytic/src.py +++ b/doc/manual/manual/contractors/analytic/src.py @@ -24,21 +24,67 @@ def tests_CtcInverse_manual(test): c = CtcInverse(f, 50) # [ctcinv-1-end] + + # # Generates the documentation figure: + + # sty = PavingStyle( + # [Color.black(),Color.black()], + # [Color.none(),Color.none()], + # [Color.none(),Color.none()]) + + # l = [ + # IntervalVector([[0,3],[2,4]]), + # IntervalVector([[0,3.5],[0,1]]), + # IntervalVector([[-2,-1],[0,3]]), + # IntervalVector([[0,2],[-3,-2]]), + # IntervalVector([[-5,-1],[-4.5,-1]]), + # IntervalVector([[3,4],[-3.5,-0.5]]) + # ] + + # fig = Figure2D("My figure", GraphicOutput.VIBES | GraphicOutput.IPE) + # fig.set_axes(axis(0,[-6,6]), axis(1,[-6,6])) + + # for xi in l: + # fig.draw_box(xi, [Color.blue(),Color.blue(0.1)]) + # c1 = CtcFixpoint(c) + # xi = c.contract(xi) + # fig.draw_box(xi, [Color.blue(),Color.white()]) + # c1 = CtcFixpoint(c,0) + # xi = c1.contract(xi) + # fig.draw_box(xi, [Color.red(),Color.none()]) + + # fig.pave([[-6,6],[-6,6]], c, 5e-3, sty) + + # [ctcinv-2-beg] - DefaultFigure.pave([[-6,6],[-6,6]], c, 1e-2) + z = IntervalVector([[0,3.5],[0,1]]) + DefaultFigure.draw_box(z, [Color.blue(),Color.blue(.1)]) # prior to contraction + z = c.contract(z) + DefaultFigure.draw_box(z, Color.blue()) # after one CtcInverse contraction + # z == [ [1.84, 3.5] ; [0, 1] ] + + # Combining CtcInverse with a CtcFixpoint: + cfix = CtcFixpoint(c) + z = c.contract(z) + DefaultFigure.draw_box(z, Color.red()) # after a fixed point contraction + # z == [ [1.84, 2.483] ; [0, 1] ] # [ctcinv-2-end] # [ctcinv-3-beg] - s = SepInverse(f, [0,50]) - DefaultFigure.pave([[-6,6],[-6,6]], s, 1e-2) + DefaultFigure.pave([[-6,6],[-6,6]], c, 1e-2) # [ctcinv-3-end] # [ctcinv-4-beg] - cu = CtcInverse(f,50) | CtcInverse(f,150) | CtcInverse(f,250) - DefaultFigure.pave([[-6,6],[-6,6]], cu, 1e-2) + s = SepInverse(f, [0,50]) + DefaultFigure.pave([[-6,6],[-6,6]], s, 1e-2) # [ctcinv-4-end] # [ctcinv-5-beg] + cu = CtcInverse(f,50) | CtcInverse(f,150) | CtcInverse(f,250) + DefaultFigure.pave([[-6,6],[-6,6]], cu, 1e-2) + # [ctcinv-5-end] + + # [ctcinv-6-beg] x = VectorVar(2) f = AnalyticFunction([x], x[0]) @@ -48,17 +94,17 @@ def tests_CtcInverse_manual(test): y = IntervalVector([[0.5,3],[-1,1]]) c.contract(y) # [[1,3],[-1,1]] # Only the first component is constrained by the not-in condition - # [ctcinv-5-end] + # [ctcinv-6-end] test.assertTrue(y == IntervalVector([[1,3],[-1,1]])) - # [ctcinv-6-beg] + # [ctcinv-7-beg] x = VectorVar(2) f = AnalyticFunction([x], x[0]-x[1]) c = CtcInverse(f, 0) assert c.fnc().input_size() == 2 assert c.fnc().output_size() == 1 - # [ctcinv-6-end] + # [ctcinv-7-end] if __name__ == '__main__': unittest.main() \ No newline at end of file From 262835ade1ca3a1052689775454bd93752441199 Mon Sep 17 00:00:00 2001 From: Simon Rohou Date: Tue, 10 Feb 2026 14:08:16 +0100 Subject: [PATCH 2/3] [doc] updated CtcInverse manual page for contraction on boxes (minor update) --- .../contractors/analytic/himmelblau_50.png | Bin 60363 -> 64480 bytes .../analytic/himmelblau_50_150_250.png | Bin 156932 -> 154495 bytes .../contractors/analytic/himmelblau_boxes.png | Bin 39850 -> 29955 bytes .../manual/contractors/analytic/src.cpp | 2 +- doc/manual/manual/contractors/analytic/src.m | 2 +- doc/manual/manual/contractors/analytic/src.py | 2 +- 6 files changed, 3 insertions(+), 3 deletions(-) diff --git a/doc/manual/manual/contractors/analytic/himmelblau_50.png b/doc/manual/manual/contractors/analytic/himmelblau_50.png index 239f44eaaf1ddb8b1d2266d21d6d3e333480617c..dfb65745b64828e9b8095a7a1228989ba13c5df7 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zGV_7F?g@Gmxba{!!rZ*x{+z3VT)dxII6LjSRXK z?=)H@-2atocDg5O%~^`b)sIJ$} zrSa-5sXt;X^1dlgG|5POycutx&0BI_%7QGDX}{f!M7wk2#qu||+x)5K_Bl(0MRUu| z``=a23N|q>-+=TZZn45L_f8(JiU-kBxiS$SZ+1R&thVB=I~py7)qaZeVW5xKxC$jV zggMCiG6n0|_Kgi+uvJ}`d#3vR&h-sL(`))a=-YaI$Yphu4L<03*N{)UHC#yO?#HLG zPguGb3E{QF6qCnp%)#S-DyIK<^`R~;rEkkIYELq112SqEd5)InV-K5WsqW8XHRIMAW?`I9cQ|g@;&k&{P}CZ2 zmB{jp(EcVpeCgcJlHW0jbaT6pr##CEoPNyk^y;-xYL-KcsP0p9`~Jcl)-anpS2ld( z?of)URTe)G{SF5Qazu!aWjVtk#_GqdC7bo>l?B#DH|@M8m`s9}rrmQ>9l5~OFKJ#^?n@W!J7J-?XOLVNr8+8)Sa;@&M?S`EVU=a8 zM0L;k6kD4cbxL=hFH6}~n3mA6^fjB%o>qvqmFNjL9BaRoYeH)LSih#bR;44Sqg3#t zf#LSN>>sg-byj~?1!}LGY`-HhhC80A`{8#oTUZ2Dx-{?GWf7(BC0xEqELxxJ-U>ch z>Q1p4O!NKYS1vHyYSCP=zl_uGq`8mJ#qz6nI=&ym()2|s=#mwi7|Ce4NZ+H_QP-M#nsSYG^JgBcY} diff --git a/doc/manual/manual/contractors/analytic/src.cpp b/doc/manual/manual/contractors/analytic/src.cpp index 5e39b87c1..2b8438005 100644 --- a/doc/manual/manual/contractors/analytic/src.cpp +++ b/doc/manual/manual/contractors/analytic/src.cpp @@ -33,7 +33,7 @@ TEST_CASE("CtcInverse - manual") IntervalVector z({{0,3.5},{0,1}}); DefaultFigure::draw_box(z, {Color::blue(),Color::blue(.1)}); // prior to contraction c.contract(z); - DefaultFigure::draw_box(z, Color::blue()); // after one CtcInverse contraction + DefaultFigure::draw_box(z, {Color::blue(),Color::white()}); // after one CtcInverse contraction // z == [ [1.84, 3.5] ; [0, 1] ] // Combining CtcInverse with a CtcFixpoint: diff --git a/doc/manual/manual/contractors/analytic/src.m b/doc/manual/manual/contractors/analytic/src.m index de2edd175..c79b5574c 100644 --- a/doc/manual/manual/contractors/analytic/src.m +++ b/doc/manual/manual/contractors/analytic/src.m @@ -20,7 +20,7 @@ z = IntervalVector({{0.0,3.5},{0.0,1.0}}); DefaultFigure().draw_box(z, {Color().blue(),Color().blue(.1)}); % prior to contraction z = c.contract(z); -DefaultFigure().draw_box(z, Color().blue()); % after one CtcInverse contraction +DefaultFigure().draw_box(z, {Color().blue(),Color().white()}); % after one CtcInverse contraction % z == [ [1.84, 3.5] ; [0, 1] ] % Combining CtcInverse with a CtcFixpoint: diff --git a/doc/manual/manual/contractors/analytic/src.py b/doc/manual/manual/contractors/analytic/src.py index 76d9f8066..7c96089cf 100644 --- a/doc/manual/manual/contractors/analytic/src.py +++ b/doc/manual/manual/contractors/analytic/src.py @@ -60,7 +60,7 @@ def tests_CtcInverse_manual(test): z = IntervalVector([[0,3.5],[0,1]]) DefaultFigure.draw_box(z, [Color.blue(),Color.blue(.1)]) # prior to contraction z = c.contract(z) - DefaultFigure.draw_box(z, Color.blue()) # after one CtcInverse contraction + DefaultFigure.draw_box(z, [Color.blue(),Color.white()]) # after one CtcInverse contraction # z == [ [1.84, 3.5] ; [0, 1] ] # Combining CtcInverse with a CtcFixpoint: From 775efae33b70c29aadd1005bb4edb87e54a126a7 Mon Sep 17 00:00:00 2001 From: Simon Rohou Date: Tue, 10 Feb 2026 16:16:22 +0100 Subject: [PATCH 3/3] [doc] updated CtcInverse manual page for contraction on boxes (minor update, Matlab) --- doc/manual/manual/contractors/analytic/ctcinverse.rst | 2 +- doc/manual/manual/contractors/analytic/src.m | 4 ++-- 2 files changed, 3 insertions(+), 3 deletions(-) diff --git a/doc/manual/manual/contractors/analytic/ctcinverse.rst b/doc/manual/manual/contractors/analytic/ctcinverse.rst index 545797153..f71b89831 100644 --- a/doc/manual/manual/contractors/analytic/ctcinverse.rst +++ b/doc/manual/manual/contractors/analytic/ctcinverse.rst @@ -94,7 +94,7 @@ To represent the vectors :math:`\mathbf{x}\in\mathbb{R}^n` consistent with the c The contractor can be used as an operator to contract a :math:`n`-d box :math:`[\mathbf{x}]`. -Note that the contraction may be fast but not minimal, depending on your analytic expression. Therefore, you can also combine a ``CtcInverse`` with a ``CtcFixpoint`` to apply the contraction procedure repeatedly until a fixpoint is reached on the same box. The following code corresponds to the next figure, which shows contraction cases in blue and their fixpoint counterparts in red. +Note that the contraction may be fast but not minimal, depending on your analytic expression. Therefore, you can also combine a ``CtcInverse`` with a ``CtcFixpoint`` to apply the contraction procedure repeatedly until a fixpoint is reached on the same box. The following code corresponds to a contraction revealed in the next figure, which shows contraction cases in blue and their fixpoint counterparts in red. .. tabs:: diff --git a/doc/manual/manual/contractors/analytic/src.m b/doc/manual/manual/contractors/analytic/src.m index c79b5574c..2bc7012e7 100644 --- a/doc/manual/manual/contractors/analytic/src.m +++ b/doc/manual/manual/contractors/analytic/src.m @@ -18,9 +18,9 @@ % [ctcinv-2-beg] z = IntervalVector({{0.0,3.5},{0.0,1.0}}); -DefaultFigure().draw_box(z, {Color().blue(),Color().blue(.1)}); % prior to contraction +DefaultFigure().draw_box(z, StyleProperties({Color().blue(),Color().blue(.1)})); % prior to contraction z = c.contract(z); -DefaultFigure().draw_box(z, {Color().blue(),Color().white()}); % after one CtcInverse contraction +DefaultFigure().draw_box(z, StyleProperties({Color().blue(),Color().white()})); % after one CtcInverse contraction % z == [ [1.84, 3.5] ; [0, 1] ] % Combining CtcInverse with a CtcFixpoint: