.. _scaled_jacobian_page:

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Scaled Jacobian
***************

The **Scaled Jacobian** mesh quality criteria is a scalar measure of the deviation from the perfect element in the geometrical sense. This measure normalizes the range of reported values between [0,1] for a normal element, the value of 1 is considered a perfect element and 0 a element with a collapsed side. Negative values are also accepted for invalid elements.

The **Scaled Jacobian** is implemented for all volumetric elements (except for polyhedrons, returning 0).

For tetrahedrons and hexahedrons, the formulas are
defined in `The Verdict Library Reference Manual [1] <https://gitlab.kitware.com/third-party/verdict/-/blob/master/SAND2007-2853p.pdf>`_.

For pyramid, the minimum scaled jacobian of the four tetrahedrons formed
in the four vertices of the pyramid base is reported.

For pentahedron, a decomposition into tetrahedron is also done.

For hexahedron prisms, the minimum scaled jacobian between two pentahedrons and one hexahedron is reported.

* Geometrically the Scaled Jacobian of a **tetrahedron** can be understood by the follow figure:

	.. image:: ../images/scaled_jacobian_tetra.png
		:align: center

	The reported Scaled Jacobian will be 1 if :math:`\alpha=45^{\circ}`

* For **hexadrons** this measure return 1 for the perfect non rotated elements:

	.. image:: ../images/scaled_jacobian_hexa.png
		:align: center

	The reported Scaled Jacobian will be 1 if :math:`\alpha=0^{\circ}` for all the vertices


*To visualize the Scaled Jacobian quality criterion in your mesh:*

.. |img| image:: ../images/scaled_jacobian.png

#. Display your mesh in the viewer.
#. Choose **Controls > Volume Controls > Scaled Jacobian** or click *"Scaled Jacobian"* button |img| of the toolbar.
   
   Your mesh will be displayed in the viewer with its elements colored according to the applied mesh quality control criterion:

	.. image:: ../images/scaled_jacobian_mesh_hexa.png
		:align: center


**See Also** a sample TUI Script of a :ref:`tui_scaled_jacobian` filter.