Unit+2.Geometry

GEOMETRY
Geometry is a form of communication to describe the environment. It is necessary to know some basic elements like[| circles], straight and curve lines and polygons. These are the tools of our [|graphic language.]

[|POLYGONS]

Polygon is the line formed by straight segments within a plane. We have particular methods to subscribe polygons in circles and a general method for all polygons.
 * Regular polygons** are polygons with equal angles and equal sides. These can be subscribed in a **circumference** when its vertices touch it.

__General Method__


 * 1) To draw an inscribed polygon using this method we divide the vertical diameter equally into as many parts as the sides of the polygon. Then using a **protractor** we draw a curve from A to B and B to A. Then place a point C outside of the circle.
 * 2) To find point D, we draw a straight line from C through the second point on the vertical diameter which continues and intersects the circumference at point D. The Distance AD is one side of the polygon.
 * 3) Measure the distance AD using the protractor. Transport this distance around the circumference to get the vertices.

__Particular Methods__

It is possible to draw regular polygons using specific methods for each one.



** GEOMETRIC TRANSFORMATIONS **

Geometric transformations are changes produced in the proportion, position or sense of direction of any form, while maintaining its own shape. In graphic language the usual transformations are:


 * Change of size through similarity or proportion.
 * Change of position through moving, turning or symmetry.

__Ways to create of similar images.__ [|Resizing]

There are two ways to create similar images:
 * __From the vertex.__ We must choose one vertex (A) and it must be linked with the rest of vertices (B,C,D…). If we want to create a figure at double size, we extend the straight lines twice the distance and then we must unite the correlating vertices (B´,C´,D´…)


 * __From the external point.__ We must join the external point with each vertex to extend the straight lines twice the distance between vertices and external point. Then we must join the new vertices.



__Ways to create similar images. Change of position__

There are three processes to create similar images without changes its size:


 * [|TRANSLATION]. It is moving the shape in one direction. From the vertices we create parallel lines in the same direction and the same distance.
 * [|ROTATION]. We move the shape around one center (AM)/centre (BR) or **rotating point**.


 * [|REFLECTION.] It is the relation of position between two figures or points of the same figure. Every point is the same distance from the **central line or axis of symmetry**.

** TANGENTS **

When two circles are touching in one point or a circle is touching a straight line in one point they are **[|tangent]**. The point where they touch is called **tangent point**.

__Circle tangents__

Two circles are tangent when they touch in one point and the radius of both circles form a straight line.



__Circle –line tangent__

One circle is tangent to a straight line when both touch in one point and the angle of the radius with the straight line forms a right angle.

** LINKS **

We call links the resulting forms produced by linking different kinds of lines. It is an application of tangents. In these cases the links of different lines are produced from the tangent point.



** SPIRALS ** [|Spiral] is a curve that spins around a point that is expanding continuously from this point.

There are diverse forms of spirals. One of the most important spirals is the spiral of Archimedes. To create this spiral we must draw twelve concentric circles. Then we divide the concentric circles into into twelve equal parts. With a little arc we join the center with the point where line cuts each circle. Then we join this point with the point where the next line cuts the next circle and so on.



Other kinds of curves similar to spirals are the **volutes**. We have two methods to create the volute:


 * __From the segment.__ We draw an arc with center at A and radius AB. Using the radius from point B to where arc AB intersects the segment we draw another arc using B as the next center extending the link. Using the new intersection point and point A as the center we draw another arc to continue the link. Continue so on alternating centers A and B and drawing arcs using the most recent intersection point.
 * =====__From the polygon__ . We extend each side of the selected polygon in the same direction. Use the protractor and center at vertex 1 draw an arc with the same distance as a side of the polygon. Change the center to vertex 2. Use the distance from the intersection point to vertex 2 as the new distance to draw the next arc. Continue changing the the center to the next vertex. Use the distances to draw the arc from the previous intersection point to the next vertex and continue the link.=====



Click[| here] to practice. [|Play] a polygon game.