Title: Theory / Unimportant / CS / Algo / Graphics / Fundamentals -Shirley Marschner{ / 02 Math / 7 Triangles
Tags: hasSeenConjDef

Name: The area of \(T\)
Definition: \(\left|\begin{array}{ll}x_b-x_a&x_c-x_a\\y_b-y_a&y_c-y_a\end{array}\right|\)
Page: 45
Chapter: 2
Section: 5.7
Variables: \(T\) a triangle with ends \((x_i,y_i)\)
Definition2: \(\frac12(\sum_{i=1}^3x_iy_{i+1}-x_iy_{i-1})\)

Name: Baricentric coordinate
Definition: \(\alpha,\beta,\gamma\), such that
Page: 45
Chapter: 2
Section: 5.7
Variables: \(a=(x_a,y_a),b=(x_b,y_b),c=(x_c,y_c)\)
\(T\) the triangle \(abc\) a triangle
Definition2: \(\alpha+\beta+\gamma=1\), and
Conjdef: Coordinates system associating to a point \(\mathbf p\),
Definition3: \(\mathbf p=\alpha a+\beta b+\gamma c\)
Applied to3: \(\beta,\gamma\) correspond to the coordinate of \(\mathbf p\), in the system with origin \(\mathbf a\), and vector \(\mathbf{b-a}\) and \(\mathbf{c-a}\). The coordinate \(\alpha\) allows to attein \(\mathbf a\) from the origin

Smaller0: \(\alpha=0\)
Greater0: \(\mathbf p\) is on the edge \([\beta,\gamma]\)
Page: 45
Chapter: 2
Section: 5.7
Variables: \(T\) a triangle with ends \(a=(x_a,y_a),b=(x_b,y_b),c=(x_c,y_c)\)
\(\mathbf p=(\alpha,\beta,\gamma)\) in Barycentric coordinate
Smaller to greater: iff

Smaller0: \(\alpha=\beta=0\)
Greater0: \(\mathbf p=\mathbf c\)
Page: 45
Chapter: 2
Section: 5.7
Variables: \(T\) a triangle with ends \(a=(x_a,y_a),b=(x_b,y_b),c=(x_c,y_c)\)
\(\mathbf p=(\alpha,\beta,\gamma)\) in Barycentric coordinate
Smaller to greater: iff

Smaller0: \(0<\alpha,\beta,\gamma<1\)
Greater0: \(\mathbf p\) belongs to the interior of the triangle
Page: 45
Chapter: 2
Section: 5.7
Variables: \(T\) a triangle with ends \(a=(x_a,y_a),b=(x_b,y_b),c=(x_c,y_c)\)
\(\mathbf p=(\alpha,\beta,\gamma)\) in Barycentric coordinate
Smaller to greater: iff