Geometry - If it's not a regular polygon, what else could it be called?

Factual Questions
1

It’s been 20 years since I studied geometry.

This is my main question: What would you call the shape gravy makes on a plate? A squiggly-edged thing. What’s that called?

Is that in the catgory of amorphous geometry?

Which category does a circle fall into? It doesn’t have any edges either–or is it considered to have one edge?

I poked around mathworld.wolfram.com and found a definition of Regular Polygon. That covers squares and triangles and other things a layman would call “geometric figures”-- things with precise lines and angles.

The Main Question Rephrased: If I am cutting shapes from paper with scissors, the neat angular pieces would be called polygons, the circles would be called… something… and the wavy-edged blobby pieces would be called… something. Please fill in the blanks.
Another question:
Imagine that a minaret and a telephone pole both cast shadows that overlap on the ground. The overlapping shadows create one odd shape that can’t be distinguished as either a minaret or a telephone pole.

How do you calculate the area of that oddly shaped shadow?

How do you calculate the area of a shadow of something with a regular shape, such as a cube?
Are all my questions from the field of geometry, or some other branch of math?

I appreciate your consideration.

2

Closed curves?

3

The shape with a squiggly outline is a Jordan Curve, which is an alternative name for a simple closed curve, that is a closed curve which is not self-intersecting. A circle is also a simple closed curve.

Calculate the position (and hence the area) of shadows is a common topic in computer graphics, so you could look at books about that. The problems can be solved using basic 3D geometry, if you can calculate the equation of a line and the intersection of a line and a plane, and know the measurements (hence the coordinates of the corners) of your buildings.

The basic method would be to calculate the line passing from the light source to a corner of the object, extend the line (in 3D) until it meets the plane you are projecting the shadow onto, find the intersection of line and plane to give one corner of the shadow, and repeat for other corners. I’m sure there are short-cuts you could use for rectangles and other shapes (e.g. the shadow of a sphere is always elliptical). If working with the sun, you can assume all the sun’s rays are parallel.

I believe projective geometry would deal with shadows more formally, but I’ve never studied it.

4

Irregular polygon?

5

[sidetrack]For the shadows to intersect in any sort of ‘X’ configuration, you would need the pole and/or the minaret to be non-vertical, or each of the shadows to be cast by a different light source (but in that case, only the area of intersection would be in full shade).

6

  1. A polygon is a closed plane figure bounded by 3 or more line segments (poly = many, in this case 3 or more). Of course your gravy shape has an edge (a closed irregular curve), but it contains no line segments, so it’s not any kind of polygon (let’s not get into the infinity thing again).

  2. A circle has one edge; it’s a closed regular curve, not a polygon.

  3. Regular polygons don’t just have “precise lines and angles;” in order to be “regular” they need to have equal edges and angles. An equilateral triangle, a square, a regular pentagon, etc. are regular polygons. Any polygon with unequal edges or angles is irregular.

  4. To calculate the area of the shadow of an object, assuming the shadow is cast on a flat surface, the easiest way is to break the shadow down into simpler shapes like rectangles and triangles, then add up the areas. But aside from direct observation, you can’t ascertain the area of the shadow of an object without knowing where it is, in relation to the light source. If the light source is the sun, the shadow is constantly changing, so there’s no hypothetical shadow without knowing what time it is.

  5. **Mangetout ** is right about the intersecting shadows.

7

If you are measuring one of these shapes on paper, for example the surface of a lake, you can use a planimeter to calculate the area.

8

Sue.

or Frank.