Geometric Structures, Scales, and Projection Principles

Posted on Mar 28, 2025 in Visual arts

Geometric Structures in Design

Regular Structures

Regular structures are distinguished by having all elements arranged uniformly and following a consistent order. Common types include:

  • Symmetric (axial)
  • Radial
  • Basic unidirectional
  • Complex unidirectional

Irregular Structures

Irregular structures are characterized by elements that are uneven and lack a regular order. The main types are:

  • Radial
  • Unidirectional
  • Complex

Modules in Patterns

A module is a shape, whether regular or irregular, that repeats to form a network or pattern. The combination of two or more modules creates compound modules.

Core Geometric Concepts

Ratio and Proportion

Ratio is the relationship between two figures that have the same shape but different sizes.

Similarity in Figures

Two figures are similar when all corresponding angles are equal, and corresponding sides are proportional and arranged in the same order. Angles of similar figures are called homologous angles.

Golden Section Construction Steps

To construct the golden section (or golden ratio division) of a segment AB:

  1. Draw the bisector of segment AB to determine its midpoint O.
  2. At end A, draw a line perpendicular to AB.
  3. Draw an arc with center A and radius AO to intersect the perpendicular line at point M.
  4. Join points M and B with a line segment.
  5. With center M and radius MA, draw an arc to intersect the line segment MB at point N.
  6. With center B and radius BN, draw an arc to intersect the original segment AB at point C. Point C divides segment AB according to the golden section (AC/CB = AB/AC).

Understanding Drawing Scales

Scale Types: Natural, Reduction, Enlargement

  • Natural Scale: Uses a 1:1 ratio. Measurements on the drawing are the same as the object’s actual measurements.
  • Reduction Scale: Measurements on the drawing are smaller than reality (e.g., 1:10, 1:50). Used for large objects.
  • Enlargement Scale: Measurements on the drawing are larger than reality (e.g., 2:1, 10:1). Used for small objects.

Graphic Scale Construction Method

For the graphic construction of scales, use the scale ratio. For instance, if using centimeters and a scale of 1:50, a real length of 100 cm would be represented by a line 2 cm long (100 cm / 50 = 2 cm). Draw a line representing a convenient scaled length and subdivide it into primary and secondary units based on the chosen scale.

Principles of Projection Drawing

Projection Center and Rays

The Projection Center (V) is the point from which all projecting rays originate. These rays pass through points on the object and intersect the picture plane.

Picture Plane Definition

The Picture Plane (or Projection Plane) is the surface where the projecting rays intersect, forming the projection (image) of the figure.

Cylindrical Projection Types

When projecting rays are parallel to each other, the projection is cylindrical. It can be:

  • Orthogonal Projection: Projecting rays are perpendicular to the picture plane.
  • Oblique Projection: Projecting rays are oblique (not perpendicular) to the picture plane.

Elements in Projection Systems

Ground Line (GL) Representation

The Ground Line (GL) represents the intersection of the horizontal and vertical projection planes. It is typically drawn as a solid line with two small parallel lines or ticks at the ends.

Horizontal Projection (Plan View)

The projection of an object onto the horizontal plane is called the horizontal projection or plan view.

Vertical Projection (Elevation)

The projection onto the vertical plane is called the vertical projection or elevation.

Lateral View (Profile View)

Projections onto a profile plane (a plane perpendicular to both the horizontal and vertical planes) are called lateral views or profile views.

Reference Lines

A reference line (or projection line) connects corresponding points of an object in different projection views (e.g., connecting a point in the plan view to the same point in the elevation view). These lines are typically perpendicular to the ground line or hinge lines between views.

Point Coordinates: Cota and Remoteness

  • Cota (Height or Elevation): The distance of a point from the horizontal projection plane (HP).
  • Remoteness (Depth or Alejamiento): The distance of a point from the vertical projection plane (VP).

Lines and Planes in Projection

Trace Points of a Line

Trace points are the points where a line intersects the principal projection planes (Horizontal Plane and Vertical Plane).

Finding the Horizontal Trace (H)

The Horizontal Trace (H) of a line is the point where the line intersects the Horizontal Plane. To find it: extend the line’s vertical projection (elevation) until it intersects the ground line (at point h’). Draw a reference line perpendicular to the GL from h’. Extend the line’s horizontal projection (plan) to intersect this reference line at point H.

Finding the Vertical Trace (V)

The Vertical Trace (V) of a line is the point where the line intersects the Vertical Plane. To find it: extend the line’s horizontal projection (plan) until it intersects the ground line (at point v). Draw a reference line perpendicular to the GL from v. Extend the line’s vertical projection (elevation) to intersect this reference line at point V.

Defining a Geometric Plane

A plane in space can be determined by:

  • Three non-collinear points (A, B, C).
  • A line and a point not on that line.
  • Two intersecting lines.
  • Two parallel lines.

A line connecting any two points within a plane lies entirely within that plane.

Traces of a Plane

The traces of a plane are the lines formed by the intersection of the plane with the principal projection planes (Horizontal Trace and Vertical Trace).