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

Thales' theorem: an angle inscribed in a semicircle is right; conversely, the locus of points seeing segment AB under 90° is the circle with diameter AB.

Golden ratio φ = (1+√5)/2 ≈ 1.618; segment division in mean and extreme ratio gives parts a and b with a/b = (a+b)/a.

Regular polygon construction: central angle = 360°/n; for a regular pentagon use the diagonal/side ratio φ ≈ 1.618.

Regular hexagon: side equals circumradius; inscribe by stepping the radius six times around the circle.

Tangent from external point P to circle O: midpoint M of OP, circle centred M through O cuts the given circle at the two tangency points.

Apollonius problem: circle tangent to three given circles has up to 8 solutions, found by reducing to line/circle tangency via homothety (dilation) centres.

Inversion: a point P maps to P' with OP·OP' = r²; circles through the inversion centre map to lines not through it.

Ellipse construction (trammel/paper-strip method): major axis 2a and minor 2b; any point satisfies PF + PF' = 2a.

Harmonic ratio (cross-ratio = −1) defines the fourth harmonic conjugate; used to construct tangents and polar lines to a conic.

First dihedral system: horizontal plane PH below, vertical plane PV behind; fold line LT is their intersection, and the vertical projection is rotated 90° about LT to lie in the drawing plane.

A point's coordinates (x, y, z) map to: horizontal projection at distance x from LT with depth y; vertical projection at distance x with height z; profile projection shows y and z.

Points on a line: if two projections of a point lie on the corresponding projections of a line, the point belongs to the line (except profile lines, which need a third projection).

True length of a segment: rotate it to parallel a projection plane (difference of coordinates method) or use the right triangle whose legs are the segment's projection and the depth/height difference.

Plane traces: a plane is defined by its intersections with PH and PV (traces); a horizontal plane has only a PV trace parallel to LT.

Intersection of two planes is a line whose projections lie at the intersections of the corresponding traces (with edge/parallel cases handled separately).

Point where a line pierces a plane: contain the line in an auxiliary projecting plane, find its intersection line with the given plane, then intersect with the original line.

Third projection (profile plane PL) resolves ambiguity for profile lines and planes perpendicular to LT; distances transfer with a 45° auxiliary line or compass arcs.

Visibility in projections is decided by comparing depths (for PH) or heights (for PV) of the competing points on the crossing edges.

Axonometric projection preserves parallelism and ratios along the three axes; isometric has equal foreshortening (≈0.816 true scale, often drawn at full scale), dimetric and trimetric use two or three different scales.

Isometric axes are at 120°; circles in the three coordinate planes become ellipses with major axis ≈1.22·d and minor ≈0.71·d (d = circle diameter).

Cabinet (perspectiva caballera): front face drawn true shape, receding axis typically at 45° with a reduction factor of 0.5; cavalier uses full scale on the receding axis.

Conical (linear) perspective: parallel lines in space converge at a vanishing point; the horizon line is the trace of the picture plane with the horizontal plane through the observer's eye.

Two-point perspective: vanishing points F and F' lie on the horizon; a horizontal line's vanishing point is found by drawing through the station point a parallel to that line, then rotating it onto the picture plane.

Height of the observer (horizon height) determines the viewpoint; objects below the horizon show their top faces, above it their undersides.

Perspective scale along a receding line is obtained by the measuring-point (puntos métricos) method or by dividing a front line and projecting to a vanishing point.

In conical perspective the image of a circle is generally a conic; circles whose plane passes through the eye project as straight lines.

ISO 5457 / UNE-EN ISO 5457 A-series formats: A0 = 1 m² with sides in ratio √2; each subsequent size halves the larger side (A4 = 210 × 297 mm).

Line hierarchy: continuous thick for visible outlines, continuous thin for dimension and extension lines, dashed for hidden edges, dash-dot thin for centre and symmetry axes, continuous thin with zigzag for break lines.

Dimensioning rules (ISO 129): dimension lines parallel to the measured distance, extension lines from the object, arrowheads at the ends, numerals above the dimension line read from bottom or right.

Diameter is prefixed Ø, radius R, spherical radius SR, square □, countersink ⌵; a dimension may be given once as reference in parentheses.

Scale = drawing length / real length; standard scales are 1:1, 1:2, 1:5, 1:10, 1:20, 1:50, 1:100 and enlargements 2:1, 5:1, 10:1.

A scale factor applied to areas is the square of the linear factor: at 1:50, an area is 1/2500 of the real area.

Title block (casillero) is placed at the bottom-right of the sheet; the ISO 7200 data fields include title, drawing number, scale, date and author.

First-angle projection (ISO-E, European) places the view of the top below the front view; third-angle (ISO-A, US) places it above.

Cutting-plane lines are thick with arrows showing viewing direction; section hatching uses thin lines at 45° with spacing chosen by the hatched area size.

Developable surfaces (prisms, cylinders, cones, pyramids) can be unrolled without distortion; doubly curved surfaces (sphere, torus) cannot be developed exactly.

Development of a right cylinder: rectangle of width 2πr and height h; an oblique cut gives a sine-curve boundary.

Development of a right cone: sector of radius equal to the generatrix g and angle α = 360°·r/g.

Truncated pyramid development: find true lengths of the lateral edges by the right-triangle method, then lay out the four trapezoidal faces in true shape.

Intersection of two cylinders of equal diameter with perpendicular axes is two ellipses; in projection it appears as straight lines crossing at the axis intersection.

Intersection curve of a cone with a plane: circle, ellipse, parabola or hyperbola depending on the plane's angle relative to the generatrices (Dandelin spheres give the foci).

Sphere: any plane section is a circle; the apparent contour in orthographic projection is a circle of the sphere's true radius.

Polyhedron Euler formula: V − E + F = 2; the five regular polyhedra are tetrahedron, cube, octahedron, dodecahedron and icosahedron.

Helix development: one turn unrolls into a right triangle with legs 2πr (base circumference) and the pitch p; the helix angle satisfies tan θ = p/(2πr).

Freehand sketching uses the box method: enclose the object in a bounding box, divide it proportionally, then refine edges and ellipses before adding detail.

Proportion checking by the pencil-sight method: hold the pencil at arm's length, align a known dimension, and transfer the ratio to the drawing.

CAD layers should separate construction lines, visible edges, hidden edges, dimensions and text; line weights are assigned by layer, not individually.

In CAD, object snaps (endpoint, midpoint, intersection, tangent, perpendicular) guarantee geometric accuracy; polar tracking enforces fixed angular increments.

Parametric constraints (coincident, parallel, tangent, equal, dimension) let a model update automatically when a driving dimension changes.

Exploded axonometric views communicate assembly order; leader lines with numbered balloons link each part to a parts list.

Technical pictograms and arrows follow ISO 3864 conventions: shape carries the message (triangle = warning, circle = prohibition, square = information).

Rendering realism depends on a consistent light source, cast and contact shadows, and a material's diffuse, specular and roughness parameters.

Additive synthesis (RGB, light) and subtractive synthesis (CMYK, pigment) produce different gamuts; a colour out of the CMYK gamut is shown as a spot or Pantone ink.

HSB/HSV describes colour by hue angle (0–360°), saturation and brightness; complementary hues are 180° apart on the wheel.

Simultaneous contrast: a colour appears shifted toward the complement of its surround (Chevreul); grey on red looks greenish, grey on green looks reddish.

Value is the relative lightness of a colour; a composition reads clearly when values are separated even if hues are close.

Texture types: visual (printed or drawn) versus tactile (relief); a texture tile must repeat without visible seams (seamless pattern).

Halftone screening converts continuous tone to dots; line screen (lpi) and dot gain determine printed tonal reproduction.

Colour harmony schemes: monochromatic, analogous (adjacent hues), complementary, split-complementary and triadic (120° spacing).

Legibility depends on contrast ratio and type size; WCAG requires at least 4.5:1 contrast for normal body text.

Design process stages: brief and research, analysis, concept generation, development and prototyping, evaluation, production and delivery.

A design brief must state objectives, target audience, constraints (budget, format, deadlines) and success criteria before any sketching begins.

Divergent then convergent thinking: generate many alternatives (mood boards, thumbnails), then select using defined criteria such as legibility, cost and brand fit.

Presentation boards follow a grid; hierarchy is built with size, weight, colour and white space so the main message is read first.

A mock-up (dieline) shows a package or editorial layout in three dimensions before production; dielines must include bleed (typically 3 mm) and fold lines.

Colour specification for production is given in CMYK values plus Pantone references; RGB files are converted with an assigned ICC profile.

Print-ready artwork requires correct resolution (300 ppi at final size), embedded fonts or outlined type, and crop marks with bleed.

Design documentation records decisions and iterations; a rationale links each formal choice to a requirement from the brief.