Extremadura PAU Technical Drawing Applied to Plastic Arts and Design II cheat sheet

Ellipse: sum of distances from any point to foci is constant 2a; major axis 2a, minor axis 2b, focal distance c = √(a² − b²).

Hyperbola: absolute difference of distances to foci is constant 2a; c² = a² + b²; asymptotes pass through center with slopes ±b/a.

Parabola: points equidistant from focus and directrix; axis perpendicular to directrix through focus; vertex midpoint between focus and directrix.

Tangent to a conic from an external point: use the polar line or the auxiliary circle; for ellipse, the tangent at point P makes equal angles with focal radii.

Tangency between two circles: external tangency when distance between centers equals sum of radii; internal when equals difference.

Tangent line to a circle at point T is perpendicular to radius OT; tangent from external point P has length √(OP² − r²).

Inscribed angle theorem: angle subtended by diameter is right; angle at center is twice angle at circumference.

Golden ratio φ = (1+√5)/2 ≈ 1.618; rectangle with sides in φ is constructible via diagonal of half-square.

Regular polygon construction: divide circle into n equal arcs; central angle = 360°/n; for pentagon use diagonal/side = φ.

Point coordinates (x, y, z) map to dihedral projections: horizontal projection (x, y), vertical projection (x, z); y is depth, z is height.

Line true length: use difference of coordinates to form right triangle; true length = √(Δx² + Δy² + Δz²) when not parallel to a plane.

Plane defined by three non-collinear points, two intersecting lines, or a line and a point; traces are intersections with projection planes.

Intersection of line and plane: pass auxiliary plane through line, find intersection line with given plane, then intersect with original line.

Intersection of two planes: find two points common to both by using auxiliary planes; the line through them is the intersection.

Visibility: a point is visible if its projection is closer to the observer than the other element at the same projection ray.

Edge view of a plane is obtained when the plane is perpendicular to a projection plane; use change of plane method.

Piercing point of line with plane: use auxiliary projecting plane containing the line; intersection with given plane gives point.

Two lines are parallel if their projections on all planes are parallel (except when both are perpendicular to a projection plane).

A line is perpendicular to a plane if it is perpendicular to two non-parallel lines in that plane; its projections are perpendicular to the traces.

Shortest distance from point to plane: perpendicular distance; use auxiliary plane perpendicular to given plane to see true length.

Distance between parallel planes: measured along common perpendicular; use edge view of planes to show distance in true length.

True shape of a plane figure: obtain by change of plane so that the plane appears parallel to the new projection plane.

Theorem of three perpendiculars: if a line in a plane is perpendicular to the projection of an oblique line, it is perpendicular to the oblique line itself.

Angle between line and plane: angle between line and its projection on the plane; found by auxiliary plane containing line perpendicular to plane.

Distance between skew lines: common perpendicular; construct by projecting one line as a point (edge view) using auxiliary planes.

Euler's formula for convex polyhedra: V − E + F = 2, where V vertices, E edges, F faces.

Regular polyhedra (Platonic solids): tetrahedron (4 triangles), cube (6 squares), octahedron (8 triangles), dodecahedron (12 pentagons), icosahedron (20 triangles).

Section of a polyhedron by a plane: intersection polygon whose vertices lie on edges; find by locating piercing points of edges with plane.

Development of a prism: unfold lateral faces as rectangles; bases attached to appropriate edges; true lengths from auxiliary views.

Development of a cylinder: rectangle of width equal to circumference (2πr) and height equal to cylinder height; bases as circles.

Development of a cone: sector of radius equal to slant height, arc length equal to base circumference; angle = 360°·r/l.

Intersection of two cylinders: use auxiliary cutting planes parallel to axes; points of intersection lie on both surfaces.

Regular pyramid: lateral faces are congruent isosceles triangles; development is a fan of triangles around apex.

ISO 5455: scale ratios; preferred scales 1:1, 1:2, 1:5, 1:10, 2:1, 5:1, 10:1; always indicate scale in title block.

Line types per ISO 128: continuous thick for visible edges, continuous thin for dimension lines, dashed for hidden edges, chain for center lines.

Dimensioning rules: dimension lines parallel to measured edge, extension lines perpendicular, arrowheads at ends, numerals above dimension line.

ISO 129: place dimensions outside the view where possible; avoid crossing dimension lines; use diameter symbol Ø and radius R.

First-angle projection (ISO): view from front placed below, top view above, left view on right; third-angle used in US.

Tolerances: ISO 286 fits; clearance fit (hole larger), interference fit (shaft larger), transition fit; indicated by letter and number.

Surface roughness: ISO 1302; Ra value in micrometers; symbol with extension line indicating machining allowance.

Title block: contains drawing number, title, scale, projection method, date, and responsible department; located bottom right.

Golden ratio φ ≈ 1.618 used in layout; divide line so that whole/segment = segment/remainder.

Rule of thirds: divide frame into 3×3 grid; place key elements at intersections for balanced composition.

Visual hierarchy: use size, contrast, color, and position to guide viewer's eye; most important element largest or highest contrast.

Modular grid: consistent column and margin system; baseline grid for typography alignment.

Freehand sketching: use construction lines, proportion by comparing lengths, and perspective with vanishing points.

Color theory: complementary colors opposite on wheel; analogous adjacent; use value contrast for legibility.

Typography: sans-serif for screen, serif for print; maintain consistent x-height and spacing; line length 45–75 characters for readability.

Technical illustration: isometric projection shows three axes at 120°; dimetric and trimetric use different foreshortening.

Project documentation set: location plan, floor plans, elevations, sections, details, and schedules; each drawn to appropriate scale.

Scale selection: 1:100 for general plans, 1:20 for details, 1:5 for construction details; ensure clarity at final reproduction size.

Bill of quantities: list materials with dimensions, quantities, and units; derived from drawings and specifications.

Construction details: show materials, fixings, joints, and tolerances; use leaders and notes rather than excessive dimensions.

Site plan: show north arrow, scale bar, boundaries, access, and existing features; orient to true north.

Presentation boards: combine orthographic drawings, perspectives, and diagrams; maintain consistent graphic language.

Problem-solving sequence: define brief, analyze constraints, generate alternatives, select, develop, and document.

Model-making: scale models use same scale as drawings; indicate scale on base; use appropriate materials for representation.