# Measurements

## Measurements

Measuring is an important part, perhaps even the origin, of geometry. Cinderella has modes for measuring distances, angles, and area. Their behavior, at least in Euclidean geometry, is relatively straightforward:

• Select two points and obtain the distance between them.
• Select two lines and obtain the angle between them.
• Select a polygon, circle, or conic and obtain its area.

For most purposes that is all you have to know. However, as usual, there are many fine details that sometime make things more difficult. If you deal with non-Euclidean geometries, their crucial defining property is a strange way to measure things. What goes on exactly is described in more detail in the section Theoretical Background. For now, it is sufficient to know that in non-Euclidean geometries measurements differ from the usual Euclidean measurements. Values of distances or angles may even become complex numbers. These values can be calculated with the theory of Cayley-Klein geometries.

The treatment of measurements in such a general way is one of the core features of Cinderella. You should not worry about the strange behavior of measurement in non-Euclidean geometries. Perhaps the best way to understand it is to play with different constructions to get a feeling for its unique aspects. Allowing people to get an intuitive feel for non-Euclidean geometries is one of the main goals of Cinderella.

Another fine point comes from the measurements of areas. Cinderella is able to measure the area of polygons and conics. In both cases there are a few things that deserve some extra attention. It is easy to define the area for a polygon that does not intersect itself. But what happens if it does? The approach chosen in Cinderella is to use a general and consistent formula for area. Areas are counted with respect to an orientation. How much a point inside the polygon contributes to the area depends on its winding number with respect to the boundary.

The area of conics is also a delicate topic. It is easy to define the area of an ellipse. But what is the area of a hyperbola? Is it infinite? Is it undefined? Is it something completely different? Cinderella chooses an algebraic approach that tries to use only one formula for all the different cases. It turns out that the area of a hyperbola is most reasonably described by a complex number. So, if you make measurements of areas of conics do not be surprised if complex numbers sometimes show up.

In particular, the measurement modes are

 Distance Angle Area

#### Remark

The measurements are shown as texts. They can be used as "text objects," so they can be dragged around and be repositioned. Consult the description of text and function for these features.