gh-36531: Add preimage/inverse_image methods to Polyhedron_base5 - #42619
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gh-36531: Add preimage/inverse_image methods to Polyhedron_base5#42619Chaitanya140904 wants to merge 2 commits into
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Documentation preview for this PR (built with commit 1eef6d6; changes) is ready! 🎉 |
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Can you rebase to lastest develop |
Adds two new methods to compute the preimage of a polyhedron
under a linear map f: R^n -> R^m:
- preimage(linear_transf, new_base_ring=None)
- inverse_image(linear_transf, new_base_ring=None) [alias]
Given Q = {y : Ay <= b} and f(x) = T*x, the preimage is
{x : (A*T)x <= b}, constructed directly from the H-representation.
Also cross-links preimage in linear_transformation's SEEALSO.
Fixes: sagemath#36531
- Fix first example output: P.preimage(A) returns QQ^2 with 1 vertex, 1 ray, 1 line (not ZZ^2 with 2 rays) - Fix projection example: matrix must have nrows == ambient_dim; use T = matrix([[1],[0]]) mapping R^1 -> R^2 instead of proj = matrix([[1,0]]) which has wrong shape - Fix Q.base_ring() expected output: returns Rational Field not Integer Ring due to base_extend normalization
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Done, rebased onto latest develop |
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This PR implements the feature requested in #36531 by adding two new methods
to
Polyhedron_base5:preimage(linear_transf, new_base_ring=None)inverse_image(linear_transf, new_base_ring=None)(alias forpreimage)What problem does this solve?
Previously, SageMath provided
linear_transformationto compute the imageof a polyhedron under a linear map
f(x) = T·x. However, there was no methodto compute the preimage (inverse image) — i.e., given a polyhedron
Q ⊆ ℝᵐand a linear mapf : ℝⁿ → ℝᵐ, compute:This fills the gap and complements the existing
ConvexSet_base.linear_transformationAPI.How it works
Given the H-representation of
Q:and a linear map
f(x) = T·xwhereTis an(m × n)matrix,substituting
y = T·xgives directly:This is a valid H-representation of a polyhedron in
ℝⁿand is constructeddirectly from the H-representation without any V-representation conversion.
Changes
src/sage/geometry/polyhedron/base5.pypreimage(linear_transf, new_base_ring=None)with:r"""docstring including a LaTeX.. MATH::block showing the preimage formula(m × n)wherem = self.ambient_dim()EXAMPLES::block with 5 examples: half-space preimage, projection, scalar dilation, empty polyhedron, alias checkTESTS::block covering: ambient dim correctness, base ring check,ValueErrorfor incompatible matrix shape, backend preservation.. SEEALSO::links tolinear_transformationandinverse_imageinverse_image(linear_transf, new_base_ring=None)as a documented alias forpreimage,with its own docstring, examples, and
.. SEEALSO::linkslinear_transformation's.. SEEALSO::to cross-linkpreimageFixes #36531