:orphan:

Examples
========

This gallery walks through every public feature of ``mathematicskit.geometry``:
convex hull, Delaunay triangulation/Voronoi diagrams, segment
intersection/point-in-polygon, polygon area/centroid, and the
Frenet-Serret frame.

Each script in this gallery is self-contained and can be run directly with
``python examples/geometry/<section>/<script>.py``.

Sections
--------

- **convex_hull** -- Graham's scan, checked against scipy's Qhull.
- **triangulation** -- Voronoi diagrams and Delaunay triangulation.
- **intersections** -- ray-casting point-in-polygon and the Jordan curve
  theorem.
- **polygon** -- Euclid's equal areas, Heron's formula, and Pick's
  theorem.
- **curves** -- curvature, arc length, and the Frenet-Serret frame.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

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Bézier curves
-------------

Curves built from control points by repeated linear interpolation.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

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    <div class="sphx-glr-thumbcontainer" tooltip="Draws a cubic Bézier curve and the construction lines of de Casteljau&#x27;s algorithm at t = 0.4: interpolate between neighbouring control points, then between the new points, until one point, the curve point, is left.">

.. only:: html

  .. image:: /api/gallery/geometry/bezier/images/thumb/sphx_glr_plot_01_de_casteljau_thumb.png
    :alt:

  :doc:`/api/gallery/geometry/bezier/plot_01_de_casteljau`

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      <div class="sphx-glr-thumbnail-title">De Casteljau's algorithm and Bézier curves</div>
    </div>


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    </div>

Convex hull
-------------

Graham's angular-sweep convex hull, checked against scipy's Qhull.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

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    <div class="sphx-glr-thumbcontainer" tooltip="Graham&#x27;s scan sorts the points by angle around the lowest point, then sweeps through them, dropping any point that would make a clockwise turn. The picture shows the angular order the sweep follows and the hull it produces; the result is checked against scipy&#x27;s Qhull-based convex_hull.">

.. only:: html

  .. image:: /api/gallery/geometry/convex_hull/images/thumb/sphx_glr_plot_01_graham_scan_thumb.png
    :alt:

  :doc:`/api/gallery/geometry/convex_hull/plot_01_graham_scan`

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      <div class="sphx-glr-thumbnail-title">Graham's scan: the convex hull by an angular sweep</div>
    </div>


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    </div>

Curves and the Frenet-Serret frame
--------------------------------------

Curvature, arc length, and the moving frame along a parametric curve.


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    <div class="sphx-glr-thumbcontainer" tooltip="A helix has constant curvature and torsion -- both confirmed here against their closed forms -- and the moving tangent/normal/binormal frame is visualized in 2D projection alongside an ellipse (whose curvature varies).">

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  .. image:: /api/gallery/geometry/curves/images/thumb/sphx_glr_plot_01_helix_frenet_frame_thumb.png
    :alt:

  :doc:`/api/gallery/geometry/curves/plot_01_helix_frenet_frame`

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      <div class="sphx-glr-thumbnail-title">The Frenet-Serret frame of a circular helix</div>
    </div>


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    </div>

Distances between shapes
------------------------

The Hausdorff distance between point sets.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Compares a circle with approximating polygons. The Hausdorff distance measures the worst-case gap between the two shapes and shrinks like 1/n^2 as the polygon gains sides; a single stray point makes it large even though every other point matches.">

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  .. image:: /api/gallery/geometry/distances/images/thumb/sphx_glr_plot_01_hausdorff_distance_thumb.png
    :alt:

  :doc:`/api/gallery/geometry/distances/plot_01_hausdorff_distance`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">The Hausdorff distance between shapes</div>
    </div>


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    </div>

Enclosing circles
-----------------

The smallest circle containing a set of points.


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    <div class="sphx-glr-thumbcontainer" tooltip="Finds the smallest circle containing a random point cloud with Welzl&#x27;s algorithm. The optimal circle always passes through two or three of the points.">

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  .. image:: /api/gallery/geometry/enclosing/images/thumb/sphx_glr_plot_01_smallest_enclosing_circle_thumb.png
    :alt:

  :doc:`/api/gallery/geometry/enclosing/plot_01_smallest_enclosing_circle`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Sylvester's problem: the smallest enclosing circle</div>
    </div>


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    </div>

Point-in-polygon
------------------------------------------------

Ray casting and the Jordan curve theorem, with crossings found by
segment intersection.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

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    <div class="sphx-glr-thumbcontainer" tooltip="A simple closed polygon splits the plane into an inside and an outside (the Jordan curve theorem). A ray cast from a point crosses the boundary an odd number of times exactly when the point is inside. The crossings are found edge by edge with segment_intersection, and their parity agrees with point_in_polygon on a concave, spiral-like polygon.">

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  .. image:: /api/gallery/geometry/intersections/images/thumb/sphx_glr_plot_01_jordan_ray_casting_thumb.png
    :alt:

  :doc:`/api/gallery/geometry/intersections/plot_01_jordan_ray_casting`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">The Jordan curve theorem and ray casting: inside or outside?</div>
    </div>


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    </div>

Polygon area
----------------------------

Euclid's equal areas, Heron's formula, and Pick's theorem, all checked
against the shoelace formula.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Euclid&#x27;s theory of area (Elements I.35-41) rests on two facts: parallelograms on the same base and between the same parallels are equal in area (I.35), and so are triangles (I.37), each triangle being half of the parallelogram (I.41). Sliding the top side along its parallel line changes every shape but, as the shoelace formula in polygon_area confirms, never the area.">

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  .. image:: /api/gallery/geometry/polygon/images/thumb/sphx_glr_plot_01_euclid_equal_areas_thumb.png
    :alt:

  :doc:`/api/gallery/geometry/polygon/plot_01_euclid_equal_areas`

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      <div class="sphx-glr-thumbnail-title">Euclid's Elements, Book I: equal areas between parallels</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Checks Heron&#x27;s formula against the shoelace formula, then shows why the arrangement matters: for a thin &quot;needle&quot; triangle the textbook form sqrt(s(s-a)(s-b)(s-c)) loses most of its digits, while Kahan&#x27;s rearrangement stays accurate.">

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  .. image:: /api/gallery/geometry/polygon/images/thumb/sphx_glr_plot_02_heron_formula_thumb.png
    :alt:

  :doc:`/api/gallery/geometry/polygon/plot_02_heron_formula`

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      <div class="sphx-glr-thumbnail-title">Heron's formula and needle-like triangles</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Counts the lattice points inside and on the boundary of polygons with integer vertices, and checks Pick&#x27;s formula A = I + B/2 - 1 against the shoelace area.">

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  .. image:: /api/gallery/geometry/polygon/images/thumb/sphx_glr_plot_03_pick_theorem_thumb.png
    :alt:

  :doc:`/api/gallery/geometry/polygon/plot_03_pick_theorem`

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      <div class="sphx-glr-thumbnail-title">Pick's theorem: area from lattice points</div>
    </div>


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    </div>

Polyhedra
---------

Vertex, edge, and face counts of convex polyhedra, and Euler's formula.


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    <div class="sphx-glr-thumbcontainer" tooltip="Counts vertices, edges, and faces of the five Platonic solids and of random convex polyhedra. Every one satisfies V - E + F = 2.">

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  .. image:: /api/gallery/geometry/polyhedra/images/thumb/sphx_glr_plot_01_euler_polyhedron_formula_thumb.png
    :alt:

  :doc:`/api/gallery/geometry/polyhedra/plot_01_euler_polyhedron_formula`

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      <div class="sphx-glr-thumbnail-title">Euler's polyhedron formula: V - E + F = 2</div>
    </div>


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    </div>

Proximity
---------

The closest pair of points by divide and conquer.


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    <div class="sphx-glr-thumbcontainer" tooltip="Finds the closest pair in random point sets by divide and conquer, checks the answer with a k-d tree, and compares the running time with the brute-force comparison of every pair.">

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  .. image:: /api/gallery/geometry/proximity/images/thumb/sphx_glr_plot_01_closest_pair_thumb.png
    :alt:

  :doc:`/api/gallery/geometry/proximity/plot_01_closest_pair`

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      <div class="sphx-glr-thumbnail-title">Shamos and Hoey: the closest pair of points</div>
    </div>


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    </div>

Line simplification
-------------------

Reducing the number of vertices of a polyline within a tolerance.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Simplifies a wiggly coastline-like curve at several tolerances. Larger tolerances keep fewer vertices, and every discarded vertex stays within the tolerance of the simplified line.">

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  .. image:: /api/gallery/geometry/simplification/images/thumb/sphx_glr_plot_01_douglas_peucker_thumb.png
    :alt:

  :doc:`/api/gallery/geometry/simplification/plot_01_douglas_peucker`

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      <div class="sphx-glr-thumbnail-title">Douglas-Peucker line simplification</div>
    </div>


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    </div>

Surfaces
--------

Gaussian and mean curvature of parametric surfaces.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Computes the Gaussian curvature K of a sphere, a cylinder, and a torus. The cylinder bends in space but has K = 0, because it can be unrolled flat without stretching, while no piece of a sphere can. Integrating K over the whole surface gives 4 pi for the sphere and 0 for the torus, as the Gauss-Bonnet theorem predicts.">

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  .. image:: /api/gallery/geometry/surfaces/images/thumb/sphx_glr_plot_01_theorema_egregium_thumb.png
    :alt:

  :doc:`/api/gallery/geometry/surfaces/plot_01_theorema_egregium`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Gauss's Theorema Egregium: curvature you can measure from inside</div>
    </div>


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    </div>

Voronoi diagrams and Delaunay triangulation
--------------------------------------------------

Nearest-site partitions and empty-circumcircle triangulations, both via
scipy's Qhull wrapper.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

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    <div class="sphx-glr-thumbcontainer" tooltip="Partitions the plane around a set of sites so that each region holds every location closer to its site than to any other. Colouring a fine grid by nearest site reproduces exactly the cells that voronoi_diagram computes, and each Voronoi vertex is equidistant from the three sites whose cells meet there.">

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  .. image:: /api/gallery/geometry/triangulation/images/thumb/sphx_glr_plot_01_voronoi_diagram_thumb.png
    :alt:

  :doc:`/api/gallery/geometry/triangulation/plot_01_voronoi_diagram`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Voronoi diagrams: every location goes to its nearest site</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Triangulates a random point set with delaunay_triangulation, checks Delaunay&#x27;s defining property (no point lies inside any triangle&#x27;s circumcircle), and shows on four points that choosing the Delaunay diagonal maximizes the smallest angle and avoids needle-like triangles.">

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  .. image:: /api/gallery/geometry/triangulation/images/thumb/sphx_glr_plot_02_delaunay_triangulation_thumb.png
    :alt:

  :doc:`/api/gallery/geometry/triangulation/plot_02_delaunay_triangulation`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Delaunay triangulation: empty circumcircles and fat triangles</div>
    </div>


.. thumbnail-parent-div-close

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    </div>


.. toctree::
   :hidden:
   :includehidden:


   /api/gallery/geometry/bezier/index.rst
   /api/gallery/geometry/convex_hull/index.rst
   /api/gallery/geometry/curves/index.rst
   /api/gallery/geometry/distances/index.rst
   /api/gallery/geometry/enclosing/index.rst
   /api/gallery/geometry/intersections/index.rst
   /api/gallery/geometry/polygon/index.rst
   /api/gallery/geometry/polyhedra/index.rst
   /api/gallery/geometry/proximity/index.rst
   /api/gallery/geometry/simplification/index.rst
   /api/gallery/geometry/surfaces/index.rst
   /api/gallery/geometry/triangulation/index.rst



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 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_