Canary Islands PAU Technical Drawing Applied to Plastic Arts & Design II cheat sheet

Ellipse: sum of focal distances is constant (PF + PF' = 2a); major axis 2a, minor axis 2b, focal distance c with a² = b² + c².

Hyperbola: absolute difference of focal distances is constant (|PF − PF'| = 2a); c² = a² + b²; asymptotes pass through the centre with slope ±b/a.

Parabola: points equidistant from focus F and directrix d; with vertex at origin and axis along x, y² = 2px where p is the focal parameter (distance F to d).

Tangent to a conic from an external point: draw the circle with diameter P–F (focus) and intersect it with the focal circle of radius 2a (ellipse/hyperbola) or with the directrix-based construction (parabola); intersection points are the tangency points.

Tangent to a circle at a point T is perpendicular to radius OT; tangent from external point P is found on the circle with diameter OP.

Tangent to a circle parallel to a given direction: draw the perpendicular from the centre to that direction; the two intersections with the circle are the tangency points.

Ovals vs. ellipses: a four-centre oval is a compass construction (arcs of circles), not a true ellipse; the ellipse is only approximated, never exactly matched, by circular arcs.

Locus of points seeing segment AB under a constant angle α: two arcs of circles through A and B, with centre on the perpendicular bisector at distance (AB/2)/tan(α/2).

Conic as section of a cone: the ratio of the plane's angle to the generatrix angle decides ellipse (plane cuts all generatrices), parabola (parallel to one generatrix), hyperbola (cuts both nappes).

Dihedral system: a point is fixed by two orthogonal projections (horizontal and vertical); the reference line LT is the intersection of the two planes.

A line is named by its two projections; a line is parallel to a plane if it is parallel to any line of that plane, and perpendicular to a plane if it is perpendicular to two non-parallel lines of the plane.

True length of a segment: rotate it to parallel the vertical or horizontal plane, or use the right triangle whose legs are the segment's projection and the difference of the endpoints' distances to the reference plane.

Plane defined by three non-collinear points, a line and a point, or two intersecting/parallel lines; its traces are its intersections with the horizontal and vertical planes.

Intersection of a line with a plane: pass an auxiliary projecting plane through the line, find its intersection line with the given plane, then the crossing point of that line with the original line.

Intersection of two planes: find two points, each obtained by intersecting both planes with the same auxiliary plane; the line through them is the intersection.

A line perpendicular to a plane: its horizontal projection is perpendicular to the horizontal trace of the plane, and its vertical projection perpendicular to the vertical trace.

Edge-view (profile) method: a plane appears as a line when projected on a plane perpendicular to it, which makes true angles and true distances directly readable.

Regular polyhedra: tetrahedron (4 triangles), cube (6 squares), octahedron (8 triangles), dodecahedron (12 pentagons), icosahedron (20 triangles); Euler's formula V − E + F = 2 holds for all convex polyhedra.

Prism and pyramid: lateral surface developed as a rectangle (prism) or as triangles/sectors (pyramid); true edge lengths are needed before developing, so use rotation or triangulation.

Cylinder lateral development is a rectangle of width 2πr and height h; cone lateral development is a circular sector of radius equal to the generatrix g and angle α = 360°·r/g.

Truncated cone development: divide the base circle into equal arcs, transfer them onto the sector, and mark the truncated generatrices; the true lengths are obtained by rotation.

Sphere: no exact plane development exists; gnomonic or loxodromic approximations are used, and the surface is treated by zones and gores.

Section of a polyhedron by a plane: intersect the plane with each face plane; the segments inside each face form the section polygon; parallel faces give parallel section edges.

Section of a cone by a plane: the section is an ellipse, parabola or hyperbola depending on the plane's angle to the axis versus the generatrix angle.

Intersection of two solids (penetration): find the points where edges/generatrices of one solid meet the surface of the other; use auxiliary planes containing the axis or parallel to it.

Axonometric systems: isometric (three equal scales, axes at 120°), dimetric (two equal scales), trimetric (three different scales); isometric drawing uses a single scale along all three axes.

Isometric projection foreshortens true lengths by √(2/3) ≈ 0.816; the isometric drawing convention ignores this and uses full scale along the axes.

Cabinet or cavaliera perspective: front face true shape, receding axis at 45° with half scale; cavaliera uses full scale on the receding axis.

Conical (linear) perspective: the picture plane cuts the visual pyramid; a point's image is where its visual ray meets the picture plane, with the station point and horizon line as references.

Principal vanishing points: parallel lines in space share a vanishing point on the horizon; lines parallel to the picture plane have no finite vanishing point (they stay parallel).

Two-point perspective: verticals remain vertical, the two horizontal directions converge to two points on the horizon; the station point's distance from the picture plane sets the cone of vision.

Height of the observer's eye equals the horizon line height on the picture plane; objects below the horizon show their top face, above it their bottom face.

Shadows in perspective: a point's shadow on the ground plane is where the ray from the light source through the point meets the plane; the light's vanishing point and its ground projection drive the construction.

Scales: reduction 1:2, 1:5, 1:10, 1:20, 1:50, 1:100; enlargement 2:1, 5:1, 10:1; the scale is a ratio of drawing length to real length, written with the drawing value first.

Dimension lines are drawn parallel to the measured edge, with arrowheads at both ends; extension lines run perpendicular from the object, and figures sit above the dimension line.

Dimensions in millimetres are written without a unit symbol; a dimension is never repeated, and each feature is dimensioned once on the view where it is clearest.

First-angle projection (European/ISO): the object sits between observer and plane, so the view placed to the right of the front view is the left-side view; third-angle (US) reverses this.

Section views: cutting plane marked with thick chain line and arrows showing viewing direction; the cut surface is hatched with thin 45° lines, and adjacent parts use different hatch directions.

Line hierarchy: thick continuous for visible edges, thin continuous for dimension and hatching, dashed for hidden edges, chain for axes and cutting planes.

Standard sheet sizes ISO A series: A0 = 1 m², each subsequent size halves the previous area; A4 = 210 × 297 mm.

Title block carries scale, sheet number, projection symbol, material and author data; the projection symbol (truncated cone) distinguishes first- from third-angle.

Golden ratio φ ≈ 1.618: a rectangle of sides a and b with a/b = (a+b)/a; used for proportional grids and page divisions.

Rule of thirds: divide the frame into a 3×3 grid; placing key elements on the intersections produces balanced, dynamic compositions.

Typographic hierarchy: size, weight, colour and spacing order information; a typical scale uses 1.5×–2× jumps between levels for clear contrast.

Legibility: x-height, stroke contrast and counter size drive readability at distance; sans-serif faces with open counters work better for signage.

Signage: viewing distance in metres × 2.5 mm gives a rough minimum cap height for legibility under normal conditions.

Grid systems: modular, column and baseline grids organise layout; the baseline grid aligns text across columns and prevents drifting baselines.

Colour and contrast: WCAG AA requires a contrast ratio of at least 4.5:1 for normal text and 3:1 for large text.

Visual weight and balance: symmetric composition conveys stability, asymmetric composition conveys dynamism; negative space is an active compositional element.

Project phases: brief and research → ideation sketches → selection → technical development → documentation → presentation; each phase produces tangible output.

Ideation sketching: thumbnail series explore form quickly; perspective sketches with construction lines and vanishing points communicate volume before technical drafting.

Technical documentation set: general arrangement drawing, exploded view, detail views with dimensions, material and finish notes, and a bill of materials.

Ergonomics drives dimensions: seat height around 43–45 cm, table height 72–75 cm, stair riser 17–18 cm with tread 28–30 cm; these figures anchor furniture and interior projects.

Modularity: design around a base module so parts combine; standard panel sizes (e.g. 1220 × 2440 mm board) constrain cutting plans and waste.

Prototype and model: scale models (1:10, 1:20) test proportion and assembly before full-size construction; mock-ups test ergonomics at 1:1.

Presentation board structure: concept statement, moodboard and references, process sketches, technical drawings, final renders, and a short rationale linking form to function.

Sustainability and materials: specify material, finish and joining method explicitly; recyclability, VOC content and end-of-life disassembly are documented alongside the technical data.