This is from Indexing the Sphere with the Hierarchical Triangular Mesh, section 4.2, equation 4.4. I don't recognize the notation. $$ \vec e ( \vartheta ) = \mathbf v _ 1 \cdot \frac { \sin ( \theta - \vartheta ) } { \sin ( \theta ) } + \mathbf v _ 2 \cdot \frac { \sin ( \vartheta ) } { \sin ( \theta ) } $$
Copying it as text from the PDF gives me other characters, so I can't even look it up that way.
- What is the cursive "$ \vartheta $" thing, and what does it mean that it "runs from" $ 0 $ to $ 1 $?
- What is the "$ e $" with an arrow over it, and what are the parentheses? Is that supposed to be read as function notation, $ e $-thing of $ \vartheta $-thing?
The purpose of this equation is to determine the number of intersections between the circle defined by the intersection of a plane (a "halfspace") and a unit sphere with a segment of a great circle (a "trixel edge") on the unit sphere. The accompanying text reads,
Any edge of a triangle is given by its end points $ \mathbf v _ 1 , \mathbf v _ 2 $. They are connected by a great circle segment. Because this is not unique, we specify that they are always connected by the shorter of the two possible great circle segment connections. For HTM nodes, this is always true because depth $ 1 $ (the largest trixel) has sides that are $ \frac 1 4 $ of a great circle. All three corners of a trixel always lie in the same half-sphere. To see whether a halfspace intersects an edge, we first parameterize the great circle segment connecting $ \mathbf v _ 1 , \mathbf v _ 2 $: $$ \vec e ( \vartheta ) = \mathbf v _ 1 \cdot \frac { \sin ( \theta - \vartheta ) } { \sin ( \theta ) } + \mathbf v _ 2 \cdot \frac { \sin ( \vartheta ) } { \sin ( \theta ) } $$ where $ \vartheta $ runs from $ 0 $ to $ \theta $, the angle between $ \mathbf v _ 1 $ and $ \mathbf v _ 2 $.
The setup can be simplified without loss of generality by restricting to the two dimensional plane spanned by unit vectors $\,\mathbf v_1\,$ and $\,\mathbf v_2\,$ where $\,\theta\,$ is the angle between $\,\mathbf v_1\,$ and $\,\mathbf v_2\,$. Thus, let $$ \mathbf v_1:=(1,0),\qquad \mathbf v_2:=(\cos(\theta),\sin(\theta)). \tag{1}$$ We are given $$ \mathbf v_3 = \vec e (\vartheta) := \mathbf v_1\cdot\frac{\sin(\theta-\vartheta)}{\sin(\theta)} + \mathbf v_2\cdot\frac{\sin(\vartheta)}{\sin(\theta)}. \tag{2} $$
Using the trigonometric identity $$ \cos(\vartheta) \sin(\theta) = \sin ( \theta - \vartheta ) + \sin(\vartheta)\cos(\theta) \tag{3} $$ we get that $$ \mathbf v_3 = (\cos(\vartheta),\sin(\vartheta)) \tag{4}$$ which is a unit vector at an angle $\,\vartheta\,$ between it and $\,\mathbf v_1\,$ which can vary from $\,0\,$ to $\,\theta\,$ itself.
Note that $\theta$ is an angle theta while $\vartheta$ is a variant form of theta and in this case is another angle.