Module 5

Tasks

Solutions | References
Table of Modules

Computation and analysis of a triangle

Algorithms in the field of computational geometry [1,2]
Consider a triangle , fig. 1.


fig. 1

Its vertices Pi shall be given by their cartesian coordinates (xi, yi):

P1 = (x1, y1),   P2 = (x2, y2),   P3 = (x3, y3).

5.1.

Based on fig. 1, develop a succinct formula (which mirrors the inherent symmetry of the problem) for the area A of the triangle depending on the coordinates xi and yi of its vertices Pi (see module 2).
Explain the basic idea and give a detailed derivation of the formula.

5.2.

Analyse the interdependance of A and the "orientation" P1->P2->P3->P1.

5.3.

Based on fig.1, develop a compact formula for the coordinates xf and yf of the crossing point F of the edge and its altitude, depending on the coordinates xi and yi of the vertices Pi (see Module 2).

Explain the basic idea and give a detailed derivation.

5.4.

Using the result of 5.3, develop a simple algorithm which allows to infere from the coordinates xi and yi of the vertices Pi:
F lies on the edge between , i.e.
on the line between P1 and P2 in front of P1
behind P2
on

The angle at P1 and P2 is acute.
P1 is obtuse, hence the triangle is obtuse.
P2
P1 is right, hence the triangle is right.
P2

5.5.

Extend the result of 5.4. to a classification algorithm:
is acute.
right.
obtuse.

5.6.

The coordinates of the ... of a triangle

can be expressed by the coordinates xi and yi of the vertices Pi using cylindrical notation (cf. [1,2] and module 2)

orthocenter H = (x h , y h)


fig. 2

(1)
(2)
circumcenter M = ( x m, ym)


fig. 3

(3)
(4)
Table 3

One half of the cells of the cylindrical numerator in eq. (1)-(4) contains variables xi or yi, the other half contains expressions like e.g. x1x3 + y1y3.

Rearrange the cylindrical numerators so that each cell contains a variabke xi or yi (see module 2, eq. (6) and (7)).

Tasks |  Solutions |  References
Table of Modules