Madrid PAU Technical Drawing Applied to Arts and Design II cheat sheet

Thales' theorem: an angle inscribed in a semicircle is right; conversely, the circumcenter of a right triangle is the midpoint of the hypotenuse.

Golden ratio φ = (1+√5)/2 ≈ 1.618; a segment divided in extreme and mean ratio satisfies a/b = (a+b)/a.

Regular polygon construction: central angle = 360°/n; for a regular pentagon the diagonal-to-side ratio equals φ.

Inversion of a point P with respect to a circle of center O and radius r: OP·OP' = r², with P' on ray OP.

Power of a point P with respect to a circle: Pow(P) = d² − r², where d = OP; tangent length from P is √Pow(P).

Radical axis of two circles is the locus of points with equal power; it is perpendicular to the line joining centers.

Apollonius problem: circle tangent to three given circles; up to eight solutions, solved by inversion or homothety.

Archimedean spiral: polar equation r = a + bθ; used for cams and decorative design motifs.

Homothety center O, ratio k: OP' = k·OP, with P' on line OP; k > 0 preserves orientation, k < 0 reverses it.

A homothety maps a circle to a circle; the ratio of radii equals |k| and the centers are homologous points.

Similarity transformation = homothety composed with an isometry; it preserves angles and ratios of lengths.

Inversion with respect to a circle: a line not through the center maps to a circle through the center; a circle through the center maps to a line.

Inversion is conformal but reverses orientation; it maps angles to equal angles.

Composition of two homotheties with centers O₁, O₂ and ratios k₁, k₂ is a homothety with ratio k₁k₂ (if k₁k₂ ≠ 1) whose center lies on O₁O₂.

In design, inversion generates curves of constant power used in gear tooth profiles and optical reflector shapes.

Tangency condition: two circles are tangent when the distance between centers equals the sum (external) or absolute difference (internal) of their radii.

Tangent line to a circle at point T is perpendicular to the radius OT.

Ellipse: sum of distances to foci is constant 2a; eccentricity e = c/a < 1; major axis 2a, minor axis 2b, with a² = b² + c².

Parabola: points equidistant from focus and directrix; standard form y² = 2px, with p = distance from focus to directrix.

Hyperbola: absolute difference of distances to foci is constant 2a; asymptotes have slopes ±b/a; e = c/a > 1.

Tangent to a conic from an external point: two real tangents if the point lies outside the conic; construction uses the polar or auxiliary circle.

Osculating circle at a point of a curve has the same tangent and curvature; its radius is the radius of curvature ρ.

Technical curves used in design: cycloid (rolling circle), involute (unwinding string, gear teeth), and catenary (hanging chain).

Monge projection: a point is given by its horizontal projection (plan) and vertical projection (elevation); the ground line separates the planes.

A line is parallel to a plane if it is parallel to any line in that plane; a line is perpendicular to a plane if it is perpendicular to two non-parallel lines in the plane.

True length of a segment is obtained by the right-triangle method: hypotenuse of a right triangle with legs equal to the segment's projection and its difference in height (or depth).

Intersection of two planes is a line; find it by locating two points common to both planes using auxiliary planes.

Distance from a point to a plane: drop a perpendicular from the point to the plane and measure its true length.

Edge-view (point view) of a line is obtained by projecting along the line direction; it is used to find true angles and distances.

Plane represented by its traces: horizontal trace (intersection with horizontal plane) and vertical trace (intersection with vertical plane).

A plane perpendicular to the ground line appears as a line in both plan and elevation.

Euler's formula for convex polyhedra: V − E + F = 2, where V = vertices, E = edges, F = faces.

Regular polyhedra (Platonic solids): tetrahedron (4 triangles), cube (6 squares), octahedron (8 triangles), dodecahedron (12 pentagons), icosahedron (20 triangles).

Prism: lateral surface area = perimeter of base × height; volume = area of base × height.

Pyramid: volume = (1/3) × area of base × height; lateral faces are triangles meeting at the apex.

Cylinder: lateral area = 2πrh; total area = 2πr(h + r); volume = πr²h.

Cone: lateral area = πrl (l = slant height); volume = (1/3)πr²h; l² = r² + h².

Sphere: surface area = 4πr²; volume = (4/3)πr³.

Developable surfaces (cylinder, cone) can be unfolded without stretching; doubly curved surfaces (sphere) cannot.

First-angle projection: the object is placed between observer and projection plane; the view from the front is placed above the top view in the layout.

Third-angle projection: the projection plane lies between observer and object; the top view is placed above the front view (common in ISO and US practice).

Dimensioning rules (ISO 129): dimension lines are parallel to the measured direction, with arrowheads at ends; extension lines start with a small gap from the object.

Each dimension is given once; avoid duplicate or redundant dimensions; place dimensions outside the view where possible.

Standard scales (ISO 5455): 1:1, 1:2, 1:5, 1:10, 1:20, 1:50, 1:100; enlargement scales 2:1, 5:1, 10:1.

Line types (ISO 128): continuous thick for visible edges, continuous thin for dimension and extension lines, dashed for hidden edges, chain line for center lines.

Tolerance and fits: ISO 286 designates holes with capital letters (H7) and shafts with lowercase (h6); clearance fit when hole tolerance zone lies above shaft.

Surface roughness is indicated by the ISO 1302 symbol with Ra value in micrometers.

Isometric projection: three axes at 120°; equal foreshortening along all three axes (scale factor ≈ 0.816 in true isometric, or full scale in isometric drawing).

Dimetric projection: two axes have equal scale; trimetric: all three scales differ.

Cabinet projection: front face true shape; receding axis at 45° with half scale; cavalier uses full scale on receding axis.

Perspective: one-point (front face parallel to picture plane), two-point (vertical edges parallel to picture plane), three-point (all axes oblique).

Horizon line = eye level; vanishing points lie on the horizon for horizontal lines.

In two-point perspective, the station point distance from the picture plane determines the cone of vision; keep the object within a 30° cone to avoid distortion.

Measuring points method: locate measuring points on the horizon to transfer true lengths along receding axes.

Shadows in perspective: light source direction determines shadow vanishing points; use them to construct cast shadows consistently.

Sketching proportions: use a pencil held at arm's length to compare relative lengths; establish a bounding box before adding detail.

Construction lines: light guide lines for axes and symmetry; darken only final visible edges to keep the sketch readable.

Orthographic sketch layout: align views (front, top, side) on a common baseline; maintain projection consistency.

Presentation drawings use line weight hierarchy: thick outlines, medium internal edges, thin construction and dimension lines.

Color and tone in design representation: warm colors advance, cool colors recede; value contrast directs attention.

Digital tools (CAD, vector editors) follow the same projection and dimensioning standards as manual drawings.

Design portfolio representation: include plans, elevations, sections, and at least one perspective or axonometric to communicate volume.

Annotation in design drawings: label materials, dimensions, and scale; include a title block with project name, scale, and date.