# Geometrica

## Bruno Autin, CERN, Switzerland

### Euclidean and Analytical Geometries

Primary concept in EUCLIDEAN geometry: POINT -> superb reasoning (unfortunately disappearing!)
Primary concept in ANALYTICAL geometry: COORDINATES -> unification of geometry and algebra (Descartes and the theory of curves)

### Browsing Geometrica

The best navigation consists of choosing a simple problem and seeing how it is solved. We shall study the generation of a parabola as the envelope of one edge of a square ruler whose vertex describes a line and whose other edge passes through a fixed point.

#### Point, line, and first drawing

The straight line is the y axis.

#### Bound point and Euclidean line

The edge passing through m is defined by a second point bound to d.

#### Geometrical function

The other edge d2 is perpendicular in m1 to d1.

#### Derivative of a geometrical function

When the parameter t varies, the line

#### Animation and story board

One can now recapitulate the construction by using the command Movie, which produces the six figures by integrating the figure n+1 to the nth one.

The movie is adapted to a demo, but to keep the trace of the construction, it is more appropriate to use StoryBoard, which puts the figures in a table.

#### Viewing the envelope process

We are now curious to see the motion of the line d2. We then make a table of the graph of the line for a whole range of t.

#### Study of a conic

To find the equation of the envelope, we elminate t between the coordinates of mp.

Geometrica can recognize the nature of the conic from its Cartesian equation
r x y + t

and determine its focus and directrix.

### New Functions

#### Parallel and paraxial lines

##### Walls as examples of parallel polygons

A line L' is parallel to a line L when it is generated by the end of a segment attached and normal to L. This defintion assumes that L represents a continuous and derivable function, which is not true for a polygon that has no normal at a vertex. A convention is introduced in Geometica to define a parallel polygon. It is illustrated for the walls of a house.

The wall has been defined with openings that can be seen if the option Ribbon is used. The inner polygons of the "ribbon" can be painted to distinguish the full and empty parts of the wall.

The drawing can finally be completed with the dimensions of the wall.

##### Paraxial rays in optics

Paraxial lines are generated in the same way as parallel lines, but the distance between the lines varies according to a prescribed function.

This is a regular motion, according to the amplitude of the oscillation; the paraxial line may take fancier shapes.