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### 2.2 Triangle reference and world space

The entire simulation domain can be represented by a set of triangles

with
triangle cell primitives C_{i} only overlapping at their boundaries. Dropping the subindex i, each cell
area is given with
and
(x_{n},y_{n}) referring to one of the three vertices of the triangle C, see Fig. 2.2 for an example. We refer to
the space of the coordinates (x,y) as world space, whereas (ξ,η) are coordinates in reference
space.

By applying affine transformations, each triangle C_{i} and the conserved quantities can be mapped
from world space to a so-called reference triangle. For the sake of clarity, the formulae in the following
sections are given relative to such a reference triangle.

We use the triangle reference space with both triangle legs of a length of 1 and aligned at the x-
and y-axes, see Fig. 2.3. The support in reference space is then given by

Using a homogeneous point representation, mappings of points from reference to world space are
achieved with

Assuming a mathematical formulation which is independent of the spatial position of the triangle, we
can drop the orientation-related components and simplify this mapping to

by
defining one of the triangle’s vertices as the world space origin. Only considering the matrix
formulation, this is more commonly known as the Jacobian
| (2.2) |

computing the derivatives in world space with respect to reference space coordinates (ξ,η).
Affine transformations to and from world space can then be accomplished by additional
projections of particular terms (see [AS98,HW08]) and for our simulations on the sphere in
Section 6.5.

With the points

parameterized with w, the interval for each edge is given by e_{i} := {e′_{i}(w)|w ∈ [0,1]}. Note the unique
anti-clockwise movements of the points e′_{i}(w) for all edges and a growing parameter w which is
important for the unique storage order of quantities at edges.
For the remainder of this chapter, we stick to the world-space coordinates.