PAU Technical Drawing II (Aragón) cheat sheet

Triangle construction given sides a, b, c: draw a, then intersect arcs of radii b (from one end) and c (from the other); the construction exists only if each side is less than the sum of the other two.

Triangle given side a and adjacent angles B, C: lay off the angles at the endpoints of a; the third vertex is where the two rays meet. Given two sides and the included angle, copy the angle first, then transfer the sides.

Circumcentre = intersection of perpendicular bisectors (equidistant from vertices); incentre = intersection of angle bisectors (equidistant from sides); orthocentre = intersection of altitudes; centroid = intersection of medians, dividing each median 2:1 from the vertex.

Regular polygon inscribed in a circle of radius R: side length s = 2R·sin(180°/n); apothem a = R·cos(180°/n). For a square s = R√2, for a regular hexagon s = R.

General method to inscribe any regular polygon in a circle: divide a diameter into n equal parts, then draw the ray from a pole (endpoint of the perpendicular diameter) through the second division point; its intersection with the circle gives the first side.

Quadrilateral constructions: a parallelogram is fixed by two sides and the included angle or by its diagonals (bisecting each other); a trapezoid needs both bases, one side and one angle; a cyclic quadrilateral satisfies opposite angles summing to 180°.

Parallelogram area = base × height; trapezoid area = (B + b)/2 × h; any polygon area can be computed by triangulating from one vertex or by the shoelace formula on vertex coordinates.

Homothety of centre O and ratio k: P' lies on ray OP with OP' = k·OP; lines map to parallel lines, angles are preserved, lengths scale by |k| and areas by k². Negative k gives a point on the opposite ray.

Similarity = homothety composed with an isometry; it preserves angles and multiplies all lengths by the same ratio, so perimeters scale by r and areas by r².

Inversion of centre O and power k: P' on ray OP with OP·OP' = k. A line not through O maps to a circle through O; a circle through O maps to a line not through O; a circle not through O maps to another circle.

In inversion, a figure and its image share the same tangent directions at corresponding points, so tangency is preserved; the centre O has no image (point at infinity).

Projective transformations map lines to lines and preserve cross-ratio, not lengths or angles; a projectivity between two lines is fixed by three pairs of corresponding points.

Perspectivity (central projection) between two lines: lines joining corresponding points are concurrent at the centre; Desargues' theorem states that two triangles in perspective from a point are in perspective from a line.

In a conic, the cross-ratio of the four points where lines through a conic point cut the conic is constant; Pascal's theorem: opposite sides of an inscribed hexagon meet in three collinear points.

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

Common tangents: two disjoint circles have 4 (2 external, 2 internal), tangent circles have 3, and one inside the other have none; external tangents meet at the external homothety centre, internal at the internal one.

Ellipse: sum of focal distances = 2a; b² = a² − c². Hyperbola: |difference of focal distances| = 2a; b² = c² − a². Parabola: distance to focus equals distance to directrix.

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

Archimedean spiral: r = a + bθ, so successive turns are separated by a constant radial distance 2πb; the tangent construction uses the instantaneous radius vector and the perpendicular to it.

Logarithmic (equiangular) spiral: r = a·e^(bθ); the angle between the radius vector and the tangent is constant, and successive radii grow in geometric progression.

Envelope of a family of lines: the curve tangent to every member of the family; e.g. the envelope of circles centred on a line with fixed radius is a pair of parallel lines, and the envelope of the family of trajectories of a rolling circle gives cycloidal curves.

Osculating circle at a point of a curve: centre lies on the normal at distance ρ = (1 + y'²)^(3/2)/|y''|, giving the radius of curvature used in evolutes.

A point is given by two projections; the third projection is obtained by transferring the distance to the reference line using the 45° line (or compass) between the auxiliary views.

A line's true length appears in any view where it is parallel to the projection plane; true length and inclination are found by the right-triangle method: hypotenuse = true length, legs = projection and difference of distances to the plane.

A plane is defined by three non-collinear points, a line and a point, or two intersecting/parallel lines; its traces are the lines where it cuts the reference planes.

A line parallel to a plane has a projection parallel to the corresponding trace; a line perpendicular to a plane projects perpendicular to the traces in both views.

Intersection of a line with a plane: pass an auxiliary projecting plane through the line, find its intersection line with the given plane, and the point where that line meets the given line is the piercing point.

Intersection of two planes: find two points, each obtained by cutting both planes with an auxiliary plane; the line through them is the intersection (its traces lie where the traces of the planes meet).

True shape of a plane figure: change of plane (auxiliary view) parallel to the plane, or rotate the plane about a trace until it coincides with the projection plane (abatement); distances are preserved in that view.

Section of a polyhedron by a plane: find the piercing points of the plane with the edges (or the intersection lines with the faces); the section polygon's vertices lie on the edges.

Section of a cone or cylinder by a plane: the conic type depends on the plane's angle to the axis - perpendicular gives a circle, oblique with angle greater than the generatrix's gives an ellipse, parallel to a generatrix a parabola, parallel to the axis (or steeper) a hyperbola.

Development of a right cylinder: rectangle of width 2πr and height h; of a right cone: circular sector of radius equal to the slant height g and angle 360°·r/g.

Development of a truncated cone or oblique cylinder uses the true lengths of the generatrices taken from an auxiliary true-length view; points are transferred by ordinates from a base line.

Intersection of two solids: find points where edges/generatrices of one solid pierce the surface of the other, plus auxiliary cutting planes for smooth surfaces; connect points lying on the same face.

Sphere: any plane section is a circle whose true size appears in a view parallel to the cutting plane; the intersection with a cylinder or cone is drawn point by point using auxiliary planes.

Polyhedron development: unfold each face about its edges into the plane, keeping edge lengths true; check that adjacent faces share the same edge length before closing the net.

ISO A-series formats: A0 = 841 × 1189 mm, area 1 m²; each subsequent size halves the larger side, so A4 = 210 × 297 mm. The ratio of sides is 1:√2.

Standard scales (ISO 5455): enlargement 2:1, 5:1, 10:1; full size 1:1; reduction 1:2, 1:5, 1:10, 1:20, 1:50, 1:100, 1:200, 1:500, 1:1000.

Dimension elements: dimension line with arrowheads, extension lines starting about 1 mm from the object and extending ~2 mm past the dimension line, and the value written above the dimension line, readable from below or from the right.

Dimensions in millimetres are written without units; diameters are prefixed Ø, radii R, spherical radii SR, squares □, and chamfers or countersinks with their own symbols.

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

Scale drawing values: real length = drawing length × scale denominator; e.g. a 60 mm segment at 1:5 represents 300 mm. Areas scale by the square of the ratio.

Do not dimension the same element twice, do not place dimensions on hidden lines, and place the smallest dimensions nearest the object with larger dimensions farther out.

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

Third-angle projection (American/ASA): the projection plane lies between the observer and the object, so the top view is above the front view and the left-side view is on the left.

A view shows two dimensions and hides the third; the front view gives width and height, the top view width and depth, the side view height and depth - the depth is transferred between top and side views.

Isometric axonometry: the three axes are at 120° and all foreshorten by 0.816; the isometric drawing uses full scale along the axes (ratio 1:1) as an accepted convention.

Dimetric and trimetric axonometries use different foreshortening on each axis; cabinet and cavalier oblique projections keep the frontal face true and reduce the receding axis by 1/2 (cabinet) or keep it full (cavalier).

In axonometric views ellipses replace circles: in isometric, the ellipse axes are in ratio √3:1 (approximately 1.22d and 0.71d for diameter d).

Hidden lines are omitted in axonometric views unless needed for clarity; centre lines and axes are drawn thin and extended slightly beyond the figure.

Sketching proportionally: use a light construction box and centre lines first, then darken visible edges; proportions are judged by comparing lengths against a chosen unit on the paper.

Standard architectural symbols: doors shown as a quarter-circle arc with the leaf, windows as double lines in the wall, stairs with an arrow showing the upward direction and numbered risers.

Industrial elements are drawn with sections and hatching at 45°: different parts of an assembly receive different hatch directions or spacings to distinguish them.

Threads: a screw is drawn with a thick line for the major diameter and a thin line (3/4 circle in end view) for the minor diameter; the thread run-out and chamfer are indicated.

Welds use the ISO 2553 reference line with an arrow and symbol above or below the line according to the side of the joint being welded.

Architectural floor plans cut the building at about 1 m above floor level, showing walls, openings and fixtures but not the roof; elevations show the exterior faces without perspective.

Tolerances and surface finish symbols (ISO 1302) are placed on the visible outline or on extension lines, with the machining allowance indicated by the symbol's position.

CAD coordinate entry: absolute (x,y), relative (@dx,dy), and polar (@length<angle); object snaps (endpoint, midpoint, intersection, perpendicular, tangent) constrain picks to geometric features.

2D editing operations: trim/extend to a boundary, offset by a distance, mirror about an axis, array in rectangular or polar patterns, fillet with radius and chamfer with two distances.

Layers organise geometry by line type, colour and lineweight; plotting controls whether a layer prints, and the current layer receives new objects.

3D modelling: extrude a closed profile along a direction, revolve a profile about an axis, sweep along a path, and loft between sections; boolean union, subtract and intersect combine solids.

A block (symbol) is a named group inserted with scale and rotation; editing the block definition updates all insertions, and attributes store text data attached to the block.

Plotting: model space is plotted at a scale, paper space layouts use viewports with their own scale; the plot style table maps object colours to pen widths.

Rendering and visual styles: wireframe, hidden, shaded and realistic; section planes and clipping boundaries reveal interior geometry without deleting it.

Sketching proportionally: use a light construction box and centre lines first, then darken visible edges; proportions are judged by comparing lengths against a chosen unit on the paper.

Standard architectural symbols: doors shown as a quarter-circle arc with the leaf, windows as double lines in the wall, stairs with an arrow showing the upward direction and numbered risers.

Industrial elements are drawn with sections and hatching at 45°: different parts of an assembly receive different hatch directions or spacings to distinguish them.

Threads: a screw is drawn with a thick line for the major diameter and a thin line (3/4 circle in end view) for the minor diameter; the thread run-out and chamfer are indicated.

Welds use the ISO 2553 reference line with an arrow and symbol above or below the line according to the side of the joint being welded.

Architectural floor plans cut the building at about 1 m above floor level, showing walls, openings and fixtures but not the roof; elevations show the exterior faces without perspective.

Tolerances and surface finish symbols (ISO 1302) are placed on the visible outline or on extension lines, with the machining allowance indicated by the symbol's position.