Geometry: Exploring Shapes and Mathematical Rigor

What Is Geometry?

Geometry is the study of shapes and their properties. The earliest known geometry book was Euclid’s Elements, which introduced mathematical rigor through the axiomatic method of definition, axiom, theorem, and proof.

A point has no dimension, a line extends infinitely in both directions, and a plane is flat. Collinear points are a pair of points whose alternate interior angles form the same angle.

Points

The most basic shapes in geometry are points, lines and planes. A point is a location with no size; it is usually represented by a dot. A line extends infinitely in both directions. It is drawn with arrowheads to indicate this fact. Two points that lie on the same line are called collinear points.

If you have distinct points A and B, they define a line that has a starting point (A) and an end point (B). A point C on the line between A and B determines a different ray with a different initial point. This is because a ray must be between the two ends of the line.

Lines

In geometry a line is a one-dimensional shape that has no width. Typically when we write a line with a pencil we think that it has a measurable width but this is not the case in geometry.

Two lines that do not intersect each other are called parallel lines. Lines that are not parallel but are perpendicular to each other are called skew lines.

If a line has only two fixed end points then it is a line segment. If a line has no ending point then it is called ray. Lines, lines segments and rays are used to define many shapes in geometry such as rectangles, triangles, quadrilaterals and kites.

Planes

Geometry deals with flat shapes such as lines, triangles & circles of two dimensions. It uses techniques of algebra and calculus for problem solving. The word geometry is derived from ancient Greek words geo’ meaning ‘Earth’ and metron’ meaning measurement.

A plane is a flat surface that extends infinity without width or thickness. Three non-collinear points determine a plane. When two planes intersect, they form a prismatic surface. If one of the planes cuts the other two in a line, then that plane is parallel to the other two. All the points on a plane are equal distances from the center of the plane.

Angles

In geometry, an angle is the figure formed by two rays that share a common endpoint. This endpoint is called the vertex of the angle. Angles can be measured in degrees or radians. An angle that is equal to 90 degrees is described as a right angle.

Right angles are used all around us in our daily lives. They appear in the corners of buildings and homes, as well as in art work. Right angles are important for engineers when building and maintaining roads, bridges, and other structures.

To measure an angle, you can use a protractor. Place the protractor against one side of a line and mark the point where the other side of the line intersects.

Polygons

The OGC Simple Features Access (SFA) standard defines a fundamental spatial data model consisting of the Geometry type and operations that operate on geometry values. Geometry values may contain optional Z and M ordinates representing elevation or measure values.

Polygons are many-sided coplanar plane figures including triangles, quadrilaterals and circles. Regular polygons have all interior angles less than 180 and all sides congruent. Hexagons have six pairs of corresponding sides and angles.

PostgreSQL assumes that geometry inputs are valid, allowing functions to work faster because they do not have to check topological correctness. This makes the geometry SELECT operation more efficient.

Surfaces

Surface geometry is the study of shapes characterized by their curvatures in three dimensions. The mathematical subject has a long history and is concerned with the geometric representation of curves in a variety of topological and differential-geometric points of view.

A parametric surface is a manifold in a topological space, generally a Euclidean plane, with rank two. That means that at every point, there is a neighborhood that is homeomorphic to an open subset of the plane by a continuous function whose derivative has rank two. Moreover, it has regularity properties under local parametrizations.

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