Basque PAU Technical Drawing II cheat sheet

Perpendicular bisector: arcs of equal radius r > half the segment from both endpoints intersect on the perpendicular bisector; every point on it is equidistant from the endpoints.

Thales circle: the locus of points seeing segment AB under 90° is the circle with diameter AB; used to construct right angles and tangents from an external point.

Regular n-gon by central angle: divide 360°/n on the circumcircle; side length s = 2R·sin(180°/n), apothem a = R·cos(180°/n).

Ellipse as locus: PF1 + PF2 = 2a, with c² = a² − b²; eccentricity e = c/a < 1. Hyperbola: |PF1 − PF2| = 2a, c² = a² + b², e > 1. Parabola: PF = distance to directrix, e = 1.

Tangent to a conic at point P: for the ellipse x²/a² + y²/b² = 1 the tangent at (x0,y0) is x·x0/a² + y·y0/b² = 1 (polar form).

Tangent circles: external tangency when center distance = r1 + r2; internal tangency when center distance = |r1 − r2|; the tangency point lies on the line of centers.

Tangent from external point P to circle (O,r): tangent length t = √(OP² − r²); the two tangent segments from P are equal.

Homothety (scale center O, ratio k): every point maps along ray OP with OP' = k·OP; parallel lines stay parallel and the center is a fixed point.

Inversion (center O, power k): OP·OP' = k; a circle not through O maps to a circle, a line not through O maps to a circle through O, and angles are preserved.

A point is defined by two projections (horizontal H and vertical V); the reference line LT is the intersection of the two planes. Distance to H = cota (vertical coordinate), distance to V = alejamiento (depth).

A line is determined by its two projections; its traces are where it meets H (traza horizontal) and V (traza vertical). A line parallel to LT has both projections parallel to LT.

True length of a segment: build a right triangle whose legs are the segment's projection and the difference of the other coordinate (difference of cotas or alejamientos); the hypotenuse is the true length.

A plane is given by two intersecting lines, two parallel lines, or its traces (intersection lines with H and V). A plane perpendicular to V projects edge-on in the vertical view.

A line lies in a plane if its traces lie on the corresponding traces of the plane; a point lies in a plane if it lies on a line of that plane.

Parallelism: two lines are parallel if their homologous projections are parallel; a line is parallel to a plane if it is parallel to some line in that plane.

Perpendicularity: a line is perpendicular to a plane if its projections are perpendicular to the corresponding traces of the plane (theorem of the three perpendiculars).

Intersection of two planes: find two points common to both (e.g. where traces cross) and join them; the line of intersection has traces on the traces of both planes.

Polyhedra are projected by projecting vertices and edges; visible edges are solid, hidden edges dashed. A prism's lateral development is a rectangle of width equal to the perimeter of the base and height equal to the prism height.

Cylinder lateral development: rectangle 2πr × h. Cone lateral development: circular sector of radius equal to the slant height g and angle α = 360°·r/g.

Section of a solid by a plane: find the intersection points of the plane with each edge (or generators); the section polygon's true shape requires a change of plane or rotation parallel to the cutting plane.

Sphere cut by a plane gives a circle; its true radius r = √(R² − d²) where d is the distance from the center to the plane.

Truncated cone development: an annular sector with outer radius g, inner radius g − h·(g/(R−r)) adjusted by similar triangles; compute by unrolling generatrices.

Intersection of a cylinder and a plane parallel to its axis is a rectangle (or two generatrices); a plane oblique to the axis gives an ellipse (or part of one).

Cone cut by a plane: parallel to base → circle; oblique with angle to axis greater than the semi-angle → ellipse; parallel to a generatrix → parabola; steeper → hyperbola.

Auxiliary views: project the solid onto a plane parallel to a face or cutting plane to obtain true magnitudes of sections and face shapes.

Change of plane (canvi de pla): a new projection plane is introduced perpendicular to one of the existing planes; the coordinate measured perpendicular to the replaced plane is preserved.

Rotating a point about an axis perpendicular to a projection plane: in that view the point moves on a circle; the other coordinate stays constant.

To find true length of a line by change of plane: choose a new plane parallel to the line; the new projection shows the true length directly.

To find true shape of a plane figure: apply two changes of plane, first making the plane edge-on (perpendicular to a new plane), then parallel to the figure.

Rotation about a horizontal axis parallel to V: the vertical projection shows the true length and the true angle with H after rotating the line parallel to V.

True angle between two planes: make their line of intersection a point (edge-on view) via successive changes of plane; the angle between the traces in that view is the dihedral angle.

Shortest distance from a point to a plane: project the point onto the plane along its normal; the true distance appears in a view where the plane is edge-on.

Distance between two skew lines: construct a plane through one line parallel to the other, then measure the distance from the second line to that plane.

Isometric axonometry: three axes at 120° on the paper; the three foreshortening coefficients are equal (0.816 in true isometric; 1 in isometric drawing/praxis).

Dimetric and trimetric axonometries use two or three different foreshortening coefficients; the axonometric axes derive from the triangle of traces.

Cavalier projection (perspectiva caballera): front face drawn true size; receding axis typically at 45° (or 30°/60°) with a reduction factor of 1/2 or 2/3.

Isometric circles appear as ellipses with major axis ≈ 1.22·d and minor axis ≈ 0.71·d (d = circle diameter); in cavalier the front face keeps true circles.

Conical perspective: the picture plane cuts the visual pyramid; a point's image is where the ray from the eye meets the picture plane. Vanishing points lie where lines parallel to the object edges through the eye meet the picture plane.

One-point perspective: front face parallel to picture plane; lines perpendicular to the picture plane converge at the principal point (center of vision).

Two-point perspective: two vanishing points on the horizon line; heights measured on a vertical measuring line in the picture plane.

Horizon line = eye level; objects above it show their underside, below it show their top. Ground line is the intersection of picture plane and ground plane.

First-angle (European) projection: the object is between observer and plane; the view from the front is placed above the top view, and the left-side view is placed on the right. Third-angle (ISO A/ANSI) reverses this arrangement.

Principal views: front (alzado), top (planta), and side (perfil). Align views: top directly below (or above) front; side at the same height as front.

Cut (corte): a cutting plane removes the near part; the cut surface is hatched with thin lines at 45°, spacing uniform, and hidden lines are omitted in the cut view unless needed.

Half section (semicorte): used for symmetric parts; half of the view is sectioned, the other half shows the exterior; the dividing line is a thin chain line (axis).

Broken-out section (sección parcial): bounded by a freehand or irregular thin line; used when only part of the interior needs showing.

Dimensioning rules: each dimension appears once; place dimensions outside the outline where possible; use arrows, dimension lines, extension lines, and the dimension value above the line (ISO).

Tolerances and fits: ISO system uses letters for holes (uppercase) and shafts (lowercase); H7/h6 is a clearance fit; diameters marked Ø, radii R, spherical SØ or SR.

Scale: drawing scale = drawing length / real length; standard scales 1:1, 1:2, 1:5, 1:10, 1:20, 1:50, 1:100; enlargement 2:1, 5:1.

Golden ratio φ = (1+√5)/2 ≈ 1.618; a rectangle with sides a, b satisfies a/b = φ when (a+b)/a = a/b.

Rule of thirds: divide the frame into a 3×3 grid; key elements placed on intersections or along lines give balanced composition.

Modular proportion: repeat a base module to generate grids, facades, and product forms; coherence comes from consistent module and scale.

Symmetry operations: reflection (mirror), rotation (n-fold), translation, and glide reflection generate ornamental and industrial patterns.

Form follows function analysis: identify primary function, secondary constraints (ergonomics, materials, manufacturing), then evaluate geometry accordingly.

Perspective and proportion in design sketches: foreshortening increases with distance; overlapping, size gradient, and atmospheric perspective convey depth.

Standard formats ISO 216: A0 = 1 m²; each subsequent size halves the area; A4 = 210 × 297 mm. Ratios are √2.

Construction order: lay out axes and reference lines first, then main outlines, then details, then dimensioning; keep line weights consistent (thick outlines, thin dimension/construction lines).

Freehand straight lines: mark endpoints, sight along the line, move from the shoulder; for long lines rotate the paper rather than the wrist.

Freehand circles: draw a light square or center cross first, then arcs; sketch ellipses in a bounding parallelogram with axes marked.

Line weights (ISO): visible outlines thick (0.5–0.7 mm), hidden edges dashed thin, center lines chain thin, dimension lines thin.

Lettering: use upright or slanted capital letters (ISO 3098), height ≥ 2.5 mm; keep consistent height and spacing.

Accuracy check: verify alignment between views, verify dimension consistency and closure, and confirm scale before final inking.

Sketching proportion: compare lengths against a reference dimension; use a pencil held at arm's length to gauge ratios before committing lines.