PAU Technical Drawing II (Andalusia) cheat sheet

Triangle construction from three sides: lay the longest side as base, strike arcs of the other two lengths from each endpoint; the intersection is the third vertex (possible only if each side is shorter than the sum of the other two).

Equilateral triangle on a given side a: height h = a·√3/2; circumradius R = a/√3; inradius r = a·(√3)/6.

Right triangle: the altitude to the hypotenuse satisfies h² = m·n (m, n being the segments it cuts), and each leg is the geometric mean of the hypotenuse and its adjacent projection (a² = c·m).

Square on diagonal d: side a = d/√2; square on side a: diagonal d = a·√2. Rectangle is fixed by two sides, or by one side plus the diagonal.

Rhombus: diagonals are perpendicular bisectors of each other; area = (D·d)/2. Given side and one diagonal, the other diagonal follows from (D/2)² + (d/2)² = a².

Regular polygon: central angle = 360°/n; interior angle = (n−2)·180°/n; exterior angle = 360°/n. Side a = 2R·sin(180°/n); apothem ap = R·cos(180°/n).

Regular pentagon: diagonal/side = golden ratio φ ≈ 1.618; construct by dividing the circumradius in mean-and-extreme ratio.

Regular hexagon: side equals circumradius (a = R); built directly by stepping the radius six times around the circle.

Regular octagon inscribed in a square of side L: corner cut x = L/(2+√2) ≈ 0.293·L; the octagon side is L(√2−1).

General method for an n-gon on a given side: bisect the side, raise the perpendicular, locate the centre at distance R = a/(2·sin(180°/n)) from each endpoint, then step the side around the circle.

Conic as locus: ellipse = points whose sum of distances to two foci is constant (2a); hyperbola = constant absolute difference (2a); parabola = equidistant from focus and directrix.

Ellipse elements: major axis 2a, minor axis 2b, focal distance c with a² = b² + c²; eccentricity e = c/a < 1.

Hyperbola: c² = a² + b²; e = c/a > 1; asymptotes pass through the centre with slope ±b/a and are the diagonals of the a×b rectangle.

Parabola: distance from focus to directrix is the parameter p; vertex is midway; the latus rectum measures 2p; e = 1.

Point construction: an ellipse point is obtained by intersecting arcs of radii r and 2a−r from the two foci; the parabola by arcs centred on focus and on directrix.

Tangent to a conic at a point P on the curve: ellipse and hyperbola - the tangent bisects the exterior angle of the focal radii; parabola - it bisects the angle between the focal radius and the line parallel to the axis.

Tangents from an external point to an ellipse: draw the focal circle (centre at one focus, radius 2a); the tangents pass through the intersections of this circle with the circle centred at the external point through the other focus.

Circle as limiting conic: intersection of a cone with a plane perpendicular to the axis; e = 0, foci coincide at the centre.

Cone-plane rule: plane angle relative to the generatrix determines the conic - perpendicular to axis gives circle, less inclined than the generatrix gives ellipse, parallel to a generatrix gives parabola, steeper gives hyperbola.

Locus of points at a given distance from a line = two parallel lines; from a point = a circle; equidistant from two points = perpendicular bisector; equidistant from two lines = the two angle bisectors.

Tangent from an external point to a circle: draw the circle with diameter OP (O centre, P point); its intersections with the given circle are the tangency points.

Tangent to a circle at a point T on it: the tangent is perpendicular to radius OT; construct the perpendicular at T.

Common external tangent to two circles: draw a circle centred at the larger centre with radius R−r, tangent from the smaller centre to it, then extend the radius line to the tangency point and draw the parallel radius in the smaller circle.

Common internal tangent: same procedure using radius R+r; the auxiliary circle is centred at the larger circle's centre with radius R+r.

Line tangent to a circle with a given direction: drop the perpendicular from the centre to the required direction; the foot on the circle gives the tangency point.

Fillet (link) between two lines forming an angle: the centre of the arc of radius r lies on the angle bisector at distance r/sin(θ/2) from the vertex; tangency points are the perpendicular feet from that centre.

Fillet between a line and a circle: offset the line by r and draw a concentric circle of radius R±r; the intersection of these loci is the arc centre.

Fillet between two circles: intersections of circles concentric with the given ones with radii R1±r and R2±r give the centre; external link uses R+r both, internal link uses R−r.

Tangency point between two tangent circles lies on the line joining their centres; for internally tangent circles it lies on the extension of that line.

Osculating and vertex tangents of conics: at an ellipse vertex the tangent is perpendicular to the major axis; at the parabola vertex it is perpendicular to the axis.

Thales theorem: a line parallel to one side of a triangle cuts the other two proportionally - AB/A'B = AC/A'C.

Fourth proportional to a, b, c: x = b·c/a, obtained by two rays from a vertex with segments a, b on one and c on the other, then a parallel.

Third proportional to a and b: x = b²/a; geometric mean of a and b: x = √(a·b), constructed by the altitude or leg theorem in a right triangle.

Golden section of segment AB: with M the midpoint and MB perpendicular of length MB, arc from M through the perpendicular end cuts AB in the golden point; the ratio is φ ≈ 1.618.

Similar figures: ratio of lengths = k, of areas = k², of volumes = k³.

Scale E = drawing length / real length; a scale 1:50 means 1 cm on paper equals 50 cm real, and areas scale by 2500.

Graphic scale construction: divide a segment into equal parts by Thales, then subdivide the first part into tenths for the fraction of the unit (transversal or decimal scale).

Inverse proportionality: xy = constant gives a hyperbola; direct proportionality y = kx gives a straight line through the origin.

Mean proportional construction for a segment m: place m and n end to end, draw the semicircle on the total, and the perpendicular at the junction gives √(m·n).

Dividing a segment into n equal parts: draw an auxiliary ray, step n equal arbitrary lengths, join the last to the segment end, and draw parallels through the other division points.

Reference system: horizontal plane (PH) and vertical plane (PV) meet at the ground line; a point is given by its horizontal projection (cota, distance to PH) and vertical projection (alejamiento, distance to PV).

Quadrant rule: a point in the first quadrant has both projections above/below the ground line on the same side; the fold line (LT) is the axis of symmetry between the two projections.

A line is defined by its two projections; its traces are where it meets PH (horizontal trace, where the vertical projection crosses LT) and PV (vertical trace).

True length of a segment: right triangle with one leg equal to the horizontal projection and the other equal to the difference of cotas (or alejamientos); the hypotenuse is the true length.

A plane is defined by two intersecting lines, two parallel lines, a line and a point, or its traces (intersections with PH and PV, meeting on LT).

Intersection of two planes: find two points common to both, e.g. by auxiliary horizontal planes cutting each plane in horizontals; join them.

Intersection of a line with a plane: pass an auxiliary projecting plane through the line, intersect it with the given plane, and the crossing point of the resulting line with the given line is the solution.

Parallelism: two lines are parallel if their homologous projections are parallel; a line is parallel to a plane if it is parallel to any line of that plane; two planes are parallel if two intersecting lines of one are parallel to two of the other.

Perpendicularity: a line is perpendicular to a plane if it is perpendicular to two lines of the plane (use the horizontal and frontal lines, whose projections give right angles in true shape).

Plane perpendicular to another: contains a line perpendicular to it; a plane perpendicular to PH projects as a line in the horizontal projection (projecting plane).

Change of plane (abatimiento): rotate the plane about its trace onto PH to obtain true shapes; the abscissa is preserved and the distance to the trace is the cota.

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

A polyhedron is drawn by projecting vertices, then edges, applying visibility: an edge is hidden when it is behind a face; hidden edges are dashed.

Sections by projecting planes: when the cutting plane is perpendicular to a projection plane, the section projects as a line and its true shape is obtained by abatiendo that plane.

Section of a prism/pyramid by an oblique plane: find the intersection points of the plane with each edge (auxiliary planes through the edges) and join them in order.

Cylinder: section by a plane parallel to the axis is a rectangle; perpendicular to the axis is a circle; oblique is an ellipse.

Cone: section parallel to the base is a circle; through the apex is a triangle; oblique cutting all generatrices is an ellipse; parallel to one generatrix is a parabola; steeper is a hyperbola.

Sphere: every plane section is a circle whose true shape appears when the cutting plane is abated; the great circle parallel to PH projects as the outline circle in the horizontal projection.

Development of a prism: unfold the lateral faces as consecutive rectangles whose widths are the true lengths of the base edges, plus the two bases.

Development of a cone: sector of radius equal to the generatrix g and angle α = 360°·r/g; the base circle is attached tangent to the arc.

Development of a cylinder: rectangle of base 2πr and height h, plus two circles.

Points on a surface of revolution are located by a parallel (circle at constant height) and a meridian; the outline (silhouette) generatrix gives the apparent contour.

Axonometric systems are defined by the three axes and their reduction coefficients; the axonometría is orthogonal if the projection direction is perpendicular to the picture plane.

Isometric perspective: three axes at 120° on the paper, equal reduction coefficient (0.816 real, simplified to 1), and the three unit segments project equal.

Isometric drawing: circles in the three coordinate planes appear as ellipses with major axis 1.22·d and minor axis 0.71·d (d = circle diameter); the major axis is perpendicular to the axis normal to the plane.

Dimetric and trimetric perspectives use two or three different coefficients; the axes are fixed by the chosen angles and the scales are read from a triangle of traces.

Cavalier perspective (perspectiva caballera): the frontal face is drawn in true shape on the picture plane; the receding axis is usually at 45° (or 30°/60°) with a reduction factor commonly 1/2.

Caballera reduction: a coefficient of 0.5 on the Y axis compensates for the apparent elongation of depth; the frontal faces keep true dimensions.

Axonometric construction of a point: given coordinates (x, y, z), step them along the three axes using the corresponding coefficients, then complete the parallelepiped.

Hidden edges in axonometry are drawn dashed; the outline of a curved solid is the envelope of the projected generatrices.

Military and cabinet perspectives: military keeps the horizontal plane in true shape and projects heights vertically; cabinet reduces the depth axis.

Choice rule: isometric for equal legibility on three faces; cavalier when one face carries the important true-shape detail.

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

Line types (ISO 128): continuous thick for visible edges, continuous thin for dimension and hatch lines, dashed for hidden edges, dash-dot thin for axes and symmetry, continuous thin with freehand ends for break lines.

Dimension elements: dimension line with arrowheads, extension lines starting 1–2 mm from the object, figures written above the line or interrupting it, all in millimetres without the unit symbol.

Dimensioning rules: never repeat a dimension, do not dimension hidden edges, place dimensions outside the view where possible, and dimension from a common datum for series of holes.

Radii prefixed R, diameters ⌀, spherical radii SR, squares □, countersinks and thickness t; angles in degrees, minutes and seconds.

Scale designation (ISO 5455): 1:1, enlargement 2:1, 5:1, reduction 1:2, 1:5, 1:10, 1:20, 1:50, 1:100; the scale is written in the title block.

Title block (ISO 7200) contains designation, drawing number, scale, projection method symbol (first- or third-angle), tolerances and responsible persons.

Sketching: proportion is preserved by comparing lengths against a chosen module; construction lines are thin and erased, final lines reinforced, and no eraser is used on the construction grid until the sketch is closed.

CAD fundamentals: model space vs paper space, object snaps (endpoint, midpoint, intersection, perpendicular, tangent), layers by line type, and plotting at a scale that matches the title block.

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