Plane and Solid Geometry: A Comprehensive Overview
Plane and Solid Geometry
Plane Geometry
Plane geometry is the branch of geometry that deals with figures whose points all lie in a plane.
Types of Plane Geometry
- Descriptive geometry: Solves problems in the geometry of space by using operations in a plane and figures that represent solids.
- Projective geometry: Studies the projection of figures onto a plane.
- Analytic geometry: Studies figures using a coordinate system and methods of mathematical analysis.
Conic Sections
A conic section is the curve formed by the intersection of a cone with a plane that does not pass through its apex. They are classified into three types:
- Ellipse: The locus of points in a plane such that the sum of the distances to two fixed points (foci) is constant.
- Parabola: The locus of points in a plane that are equidistant from a fixed point (focus) and a fixed line (directrix).
- Hyperbola: The locus of points in a plane for which the difference of the distances to two fixed points (foci) is constant and less than the distance between the foci.
Basic Geometric Elements
- Point: A dimensionless geometric element that describes a position in space, determined by a coordinate system.
- Straight line: A continuous and indefinite succession of points in a single dimension.
- Ray: One of the two parts into which a line is divided by any of its points.
Polygons
A polygon is a geometric figure formed by consecutive non-collinear segments, called sides. Polygons can be classified as:
- Simple: Two non-consecutive edges do not intersect.
- Complex: Two non-consecutive edges intersect.
- Convex: A line passing through it intersects it at a maximum of two points.
- Concave: A line can intersect it at more than two points.
- Regular: All angles and sides are equal.
- Irregular: Angles and sides are not equal.
- Equilateral: All sides are equal.
- Equiangular: All angles are equal.
Specific Polygons
- Triangle: A polygon with three sides.
- Cuboid: A polyhedron with six faces (hexahedron), where all faces are parallelograms, parallel and equal in pairs.
- Parallelepiped: A polyhedron with six faces (hexahedron), where all faces are parallelograms, parallel and equal in pairs.
Other Geometric Elements
- Polygon line: An ordered set of segments such that the end of one coincides with the beginning of the next. A closed polygonal line forms a polygon.
- Bisector of a segment: The line perpendicular to the segment at its midpoint.
- Bisector of an angle: The line that divides an angle into two equal parts; the locus of points equidistant from the rays of an angle.
Solid Geometry
Solid geometry, also known as geometry of space, considers figures whose points are not all on the same plane.
Polyhedra
A polyhedron is a solid figure whose faces are flat and enclose a finite volume. A convex polyhedron is one where any line segment joining two points inside the polyhedron lies entirely inside the polyhedron.
- Prism: A polyhedron with two equal and parallel faces (bases) and lateral faces that are parallelograms.
Surfaces of Revolution
- Hyperboloid: The surface generated by rotating a hyperbola about one of its axes.
- Sphere: A solid bounded by a curved surface whose points are equidistant from an interior point (center).
- Cone: A solid generated by rotating a right triangle about one of its legs.
- Cylinder: A surface formed by the points at a fixed distance from a given line (axis). It can be generated by rotating a line around a fixed axis.
Curves Generated by Circles
- Cycloid: The curve generated by a point on a circle rolling along a straight line.
- Epicycloid: The curve generated by a point on a circle rolling along the outside of another circle.
- Hypocycloid: The curve generated by a point on a circle rolling along the inside of another circle.
- Trochoid: The curve generated by a point on a circle rolling along a straight line, but not on the circumference.
- Epitrochoid: The curve generated by a point on a circle rolling along the outside of another circle, but not on the circumference.
- Hypotrochoid: The curve generated by a point on a circle rolling along the inside of another circle, but not on the circumference.
Planes and Projections
- Plane: An ideal entity with two dimensions, containing infinite points and lines.
- Half-plane: One of the two parts into which a plane is divided by a line.
- Isometric projection: A cylindrical orthogonal axonometric projection.
- Cavalier projection: An oblique parallel projection where the dimensions of the front projecting plane and elements parallel to it are true to size.
- Conic projection: A projection where the projecting lines converge at a point (observer’s eye), projecting the image onto an auxiliary plane.
- Axonometric projection: A graphical representation system using orthogonal projection onto three orthogonal axes, preserving proportions in height, width, and length. It differs from conic projection because the scale is independent of distance from the observer.
- Orthogonal projection: A projection where the projecting lines are perpendicular to the projection plane.
Angles
- Vertex: The point where two rays forming an angle meet.
- Angle: The part of a plane between two rays with the same origin. Measured in radians, degrees, or grads.
- Dihedral angle: The angle between two half-planes with a common line of intersection.
- Solid angle: The angle subtended by an object at a given point, measuring its apparent size.
Geometric Centers
- Circumcenter: The intersection of the three perpendicular bisectors of a triangle; the center of the circumcircle.
Drawing Instruments
Compass
A compass is used to draw circles and arcs. It has two legs joined by a hinge. Types of compasses include:
- Stroke compass: Has two adjustable legs with pointed ends, one with a lead.
- Beam compass: Has a fixed radius, adjusted with a screw. Used for small circles and circles with equal radii.
- Ballerina compass: Used for drawing very small circles.
- Golden compass: A three-pronged compass for dividing a line in the golden ratio.
Protractor
A protractor measures angles in degrees. Types include:
- Semicircular protractor: Graduated in 180 degrees or 200 grads.
- Circular protractor: Graduated in 360 degrees or 400 grads.
Set Squares
- Set square (45-45-90): A right triangle with a 90° angle and two 45° angles. Used for drawing parallel and perpendicular lines.
- Set square (30-60-90): A scalene triangle with angles of 90°, 60°, and 30°. Used for drawing parallel lines and specific angles.
Rulers
- Ruler: A measuring instrument with a rectangular plate and a scale divided into units of length (e.g., cm or inches).
- Scale ruler: A special ruler with a prismatic cross-section containing different scales. Used for measuring drawings with multiple scales.
Other Drawing Tools
- Pencil: A writing or drawing instrument with a thin stick of pigment (graphite, charcoal, or colored pigment) encased in wood, paper, or plastic.
- Technical pen: A drawing tool with a liquid ink reservoir. Ink is fed to the nib through a channel by gravity and capillarity.