PAU Technical Drawing Applied to Plastic Arts (Cantabria) cheat sheet

Thales' theorem: a line parallel to one side of a triangle cuts the other two sides in proportional segments (AB/BC = A'B'/B'C').

Golden section of segment AB: point P divides AB so AB/AP = AP/PB ≈ 1.618; construct with midpoint, perpendicular of length AB/2, and arc from the hypotenuse.

Regular polygon inscribed in a circle of radius R: side s = 2R·sin(180°/n); apothem a = R·cos(180°/n).

Any angle inscribed in a semicircle is 90° (Thales' circle); the locus of points seeing segment AB under angle α is an arc of constant inscribed angle.

Circle tangent to two lines: its centre lies on one of the angle bisectors; tangent to a line and a circle: centre lies on a parabola with the line as directrix.

Golden rectangle: removing a square leaves a similar rectangle; successive squares generate the logarithmic spiral used in composition grids.

Radical axis of two circles is perpendicular to the line of centres; tangent length from external point P is √(d² − r²).

Root-rectangle ratios: A-series sheets keep h/w = √2 ≈ 1.414, so halving the long side preserves the ratio.

Harmonic construction divides a segment in a given ratio internally and externally: AP/PB = AQ/QB with P and Q conjugate points.

First-angle (European) system: the object lies between observer and plane, so the top view is placed below the front view and the left view to the right.

Third-angle (ISO-A/US) system: the projection plane lies between observer and object, so the top view goes above the front view.

A point is fixed by two projections; its distance to the ground line in the vertical projection equals its height, and in the horizontal projection equals its depth.

A line parallel to a projection plane shows true length in that view; a line parallel to both horizontal and vertical planes appears true length in both.

True length of a segment from its projections: L = √(Δx² + Δy² + Δz²), combining the horizontal and vertical coordinate differences.

A plane appears as an edge (line) in the view whose projection direction lies in the plane; it appears true shape only in a view perpendicular to it.

Visibility in multiview: a point is hidden when another point with greater depth (or height) projects onto the same location; hidden edges are drawn dashed.

Intersection of two planes is a line found by locating two points common to both, usually via auxiliary cutting planes.

Change-of-plane method: a new projection plane parallel to a line or plane yields true length or true shape in one step.

Isometric axonometry: three axes at 120°, foreshortening equal on all three; along-axis lengths are drawn at 0.816 of true, or full scale in the simplified isometric.

Dimetric projection: two axes share a scale factor and the third differs; standard dimetric uses about 7° and 42° from the horizontal.

Cavalier perspective (oblique): frontal face drawn true shape, receding axis usually at 45° with depth reduced to 1/2 (cabinet) or full scale (cavalier).

Conical perspective: objects of equal size subtend smaller angles as distance grows, so parallel receding lines converge at a vanishing point on the horizon.

One-point perspective: front face parallel to the picture plane; two-point: vertical edges stay vertical and two vanishing points lie on the horizon line.

Vanishing point of a family of parallel lines: intersection of the picture plane with the line through the station point parallel to that family.

Height of an object at depth d in perspective: h' = h · (d0 / d), where d0 is the distance from the picture plane to the station point.

Circle in perspective projects as an ellipse; its major axis is not generally the horizontal diameter except when the circle lies in a horizontal plane below eye level.

Shadows in axonometry and perspective: light direction fixes the shadow of each vertex on the ground plane, and the shadow line joins those points.

ISO A-series sheet: A0 = 1 m², ratio √2; each size halves the previous area (A1 594×841 mm, A2 420×594 mm, A3 297×420 mm, A4 210×297 mm).

Scale is drawing length divided by real length; 1:2 is reduction, 2:1 enlargement, and 1:1 full size. Graphic scale bars survive photocopying, numerical ratios do not.

Dimension lines are drawn parallel to the measured edge, terminated by arrows or oblique strokes, with the numeral placed above the line and read from the bottom or right.

Dimension figures state the true size regardless of drawing scale; never scale a drawing to obtain a measurement.

Line hierarchy: continuous thick for visible outlines, dashed for hidden edges, chain thin for axes and symmetry, continuous thin for dimension and extension lines.

Extension lines start about 1 mm from the object and extend 2–3 mm beyond the dimension line; dimension lines are placed outside the outline when space is tight.

Radii are marked R, diameters ⌀, squares □, and spherical radii SR; counts and repetitions use n× before the value.

First-angle symbol: a truncated cone with the small end toward the front view; third-angle symbol shows the large end toward the front view.

Tolerance and fit notation: nominal size followed by ISO code, e.g. 40 H7/g6, where uppercase denotes the hole and lowercase the shaft.

Rule of thirds: key elements placed on the lines dividing the frame into nine equal parts, intersections acting as focal anchors.

Dynamic symmetry: root rectangles (√2, √3, √5) subdivide into proportional modules and give repeating ratios across a composition.

Gestalt grouping laws: proximity, similarity, continuity, closure and common fate determine how separate marks read as one figure.

Figure–ground contrast depends on luminance difference; equal-value adjacent areas merge and lose their edge.

Visual weight increases with size, saturation, isolation and placement high or right of centre in a left-to-right reading culture.

Linear perspective cues: relative size, overlap, texture gradient, atmospheric fading and convergence of parallels all signal depth.

Modulor and modular grids: a repeating unit sets all dimensions, so proportion is preserved when the design is scaled.

Symmetry types: reflection, rotation (n-fold), translation and glide; a composition may combine several without being fully symmetric.

Colour and value hierarchy: the highest-contrast area attracts the eye first, so focal emphasis is achieved by contrast, not by size alone.

A product drawing set typically contains general arrangement, exploded view, detail views and a parts list keyed by balloon numbers.

Exploded axonometry uses dashed assembly axes to show the order in which parts fit; each part keeps its axonometric orientation.

For an artwork or installation, specify overall envelope dimensions, mounting points, weight and material, not only the visual appearance.

Site plan scale is usually 1:100 or 1:200; furniture details 1:5 or 1:10; a single project may combine several scales, each labelled.

Section and elevation views communicate spatial experience: ceiling height, sightlines and viewer position are dimensioned from a fixed datum (finished floor level).

Material callouts use leader lines with a dot on the surface; hatches denote sectioned material and must be consistent across all views.

Prototype documentation: record the fabrication sequence, joint details and tolerances so the object can be rebuilt without the original maker.

Presentation layout follows a reading path: overview first, then plans, sections, details; scale bars and north arrows anchor every sheet.

Technical drawing communicates unambiguously through standard views, dimensions and symbols; rendering communicates atmosphere and is not dimensioned.

A dimensioned orthographic set is the legal and contractual record; perspective renderings are illustrative and not measurable.

Drawing conventions for design portfolios: consistent line weights, a single scale bar per sheet and a stated projection system avoid misreading.

Annotations should be placed outside the object outline and connected by leaders; overlapping dimension text is a communication error.

Digital exchange uses vector PDF or DWG with layers named by content (walls, dimensions, text) so others can edit without redrawing.

Colour coding in diagrams (circulation, structure, services) must be explained in a legend; colour alone cannot carry meaning for colour-blind readers.

A scale model at 1:20 or 1:50 communicates volume and spatial sequence where a flat drawing cannot; state the scale on the base.

Version control: number revisions on the sheet, keep a revision cloud and a date, so the latest drawing can be identified at a glance.