Geometric-Design

Geometric Design Overview

  • Geometric Design (GD):

    • A branch of computational geometry.

    • Involves construction and representation of free-form curves, surfaces, or volumes.

    • Closely related to geometric modeling.

    • Applications: shipbuilding, aircraft, automotive industries, and architectural design.

    • Models can be created for objects of any dimension in any geometric space (2D and 3D).

    • Important in computer graphics (2D for typography, technical drawing; 3D for CAD and medical imaging).

Historical Perspective

  • Fascination with Geometric Shapes:

    • Mathematicians have historically constructed ideal representations of geometric shapes.

    • Development of methods to measure lengths (1D), areas (2D), and volumes (3D).

Recognizing and Analyzing Geometric Shapes

Types of Geometric Design

  • Circle:

    • Defined by a curve at a uniform distance from the center.

    • The distance around the circle is the circumference.

  • Triangle:

    • Formed by 3 straight lines (sides).

    • Classification:

      • By Angles:

        • Right Triangle: One right angle.

        • Acute Triangle: All angles < 90 degrees.

        • Obtuse Triangle: One angle > 90 degrees.

      • By Sides:

        • Equilateral: All sides equal.

        • Isosceles: Two sides equal.

        • Scalene: No sides equal.

  • Rectangle:

    • A four-sided shape with all angles measuring 90 degrees.

  • Rhombus:

    • Four sides of equal length; angles may not be 90 degrees.

  • Square:

    • A special rectangle and rhombus with both equal side lengths and 90-degree angles.

  • Trapezoid:

    • A four-sided figure with only two parallel sides.

Polygons

  • Definition:

    • A polygon is a closed plane figure with straight sides.

    • Comprised of edges and vertices (corners).

Types of Polygons

  • Simple Polygon:

    • One boundary, no intersections.

  • Complex Polygon:

    • Intersecting sides.

  • Regular Polygon:

    • All sides and angles equal.

  • Irregular Polygon:

    • Varying side lengths and angle measures.

  • Concave Polygon:

    • At least one internal angle > 180 degrees.

  • Convex Polygon:

    • All angles < 180 degrees.

Angle Properties of Polygons

  • Sum of Interior Angles Formula:

    • Sum = (n-2) x 180°, where n = number of sides.

Examples: Sum of Internal Angles

  • Triangle (3 sides): 180°

  • Quadrilateral (4 sides): 360°

  • Pentagon (5 sides): 540°

  • Hexagon (6 sides): 720°(And so on for additional polygons up to Decagon.)

Solids

  • Definition:

    • Three-dimensional objects, described by dimensions of width, depth, and height.

    • Properties: volume and surface area.

Types of Solids

  • Polyhedra:

    • Solid with flat surfaces made of polygons.

    • Includes forms like prisms and pyramids.

  • Non-Polyhedra:

    • Curved surfaces (spheres, cylinders, cones, torus).

Polyhedra Examples: Platonic Solids

  • Five Platonic Solids:

    • Cube, Tetrahedron, Dodecahedron, Octahedron, Icosahedron.

Prisms

  • Definition:

    • Polyhedra with flat sides and two identical polygonal bases.

    • Classified into Regular and Irregular Prisms.

Pyramids

  • Defined by a base and an apex (vertex).

Non-Polyhedra Examples

  • Cylinder: Curved surfaces and two equal circular bases.

  • Sphere: Round, symmetrical, no edges or vertices.

  • Cone: Rotated triangle with a flat base and one curved side.

  • Torus: Formed by revolving a circle around an axis without touching.

Transformations

  • Definition: Movement of geometric figures in the coordinate plane.

Categories of Transformations

  • Rigid Transformation:

    • Shape and size remain unchanged.

  • Non-Rigid Transformation:

    • Size may change, but shape remains the same.

Types of Transformations

  • Translation: Slide without rotation or resizing.

  • Rotation: Turns a shape around a fixed point.

  • Reflection: Produces a mirror image across a line of symmetry.

  • Dilation: Resizes the shape around a center point with a scale factor.

Patterns and Diagrams

  • Geometric Patterns: Repeating geometric shapes typically used in design and architecture.

  • Symmetry Types: Reflectional, rotational, translational.

  • Rosette Patterns and Symmetry Groups:

    • Examples of symmetry groups, including dihedral and cyclic.

Cultural Aspects

  • Geometric Design in Art and Culture:

    • Influences and inspirations from nature and the environment, particularly in tribal designs in regions like Mindanao.

  • Mindanao Tribal Groups:

    • Rich traditions in weaving and design, with themes rooted in ancestry and cultural heritage.