La Rioja PAU Technical Drawing Applied to Plastic Arts and Design II cheat sheet

Thales' theorem: an angle inscribed in a semicircle is right, so to draw a right angle at P on a line, draw the circle with diameter AB where P lies on the circumference.

Perpendicular bisector of a segment is the locus of points equidistant from both endpoints; the circumcentre is the intersection of the three perpendicular bisectors.

Bisecting an angle: with centre at the vertex draw an arc cutting both sides, then intersect two equal-radius arcs from those cut points; the line through the vertex and the intersection is the bisector.

Regular polygon construction: exterior angle = 360°/n, interior angle = 180° − 360°/n; a regular n-gon is constructible with compass and straightedge only when n is a product of a power of 2 and distinct Fermat primes.

Golden ratio φ = (1+√5)/2 ≈ 1.618; a golden rectangle has sides in ratio φ, and removing a square leaves a smaller golden rectangle.

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

Radical axis of two circles is the locus of points with equal power; it is perpendicular to the line of centres and passes through the intersection points of the two circles when they meet.

Conic definition by focus and directrix: parabola e = 1, ellipse e < 1, hyperbola e > 1, where e is the ratio of distance to focus over distance to directrix.

Similar figures: linear ratio k, area ratio k², volume ratio k³; doubling a model's linear dimensions multiplies surface by 4 and volume by 8.

Scale factor = drawing length / real length; a 1:50 plan means 1 cm on paper equals 50 cm = 0.5 m in reality.

To convert between scales, multiply by the inverse ratio: a length at 1:20 becomes 1:50 by multiplying by 20/50 = 0.4.

Dividing a segment into n equal parts uses Thales: draw an auxiliary ray from one endpoint, mark n equal steps, join the last to the other endpoint, then draw parallels.

Fourth proportional to a, b, c solves a/b = c/x, so x = bc/a; the third proportional to a and b solves a/b = b/x, so x = b²/a.

Standard paper ISO 216 A series: aspect ratio √2:1; A0 area 1 m², and each successive size halves the area (A4 = 1/16 m²).

Standardisation in technical drawing fixes sheet sizes, line types, line widths and lettering so drawings are interchangeable between offices and countries.

True length of an oblique segment appears only in a view parallel to it; use change of plane or rotation to bring it parallel to a projection plane.

Angle between a line and a plane is measured in a plane containing the line and perpendicular to the given plane, seen in true size.

Dihedral angle between two planes is measured by a plane perpendicular to their line of intersection; the angle appears true size in that perpendicular section.

Distance from a point to a plane equals the perpendicular segment; in multiview it is found by projecting the point onto the plane's edge view.

Rotation method: rotate the object about an axis perpendicular to a projection plane until the element is parallel, keeping one coordinate constant.

Change-of-plane method: introduce a new projection plane parallel to the element to obtain true length, then a second change to obtain true shape.

Shortest path on a surface (e.g. a cone or cylinder) is found by developing the surface into a plane and drawing a straight line between the two points.

First-angle (European) projection places the view of the object seen from above below the front view; third-angle (American) places it above.

Number of principal views: front, top, and side elevations plus any auxiliary view needed to show inclined faces in true shape.

Hidden edges are drawn with dashed lines; centre lines with long-short dashes; visible outlines with continuous thick lines.

Dimensioning rules: each dimension appears once, on the view that shows the feature in true shape; avoid duplicating dimensions across views.

Dimension lines are placed outside the outline, parallel to the measured edge, with arrowheads touching the extension lines; values are written above the line, readable from the bottom or right.

Reference/datum dimensioning measures all dimensions from a common origin, reducing accumulated tolerance error compared with chain dimensioning.

Auxiliary view is projected perpendicular to an inclined surface so that surface appears in true size; its depth is transferred from the adjacent principal view.

Isometric projection: three axes at 120°, foreshortening factor 0.816; isometric drawing uses full scale along the axes (factor 1), giving a slightly enlarged but standard representation.

Dimetric and trimetric axonometrics use different foreshortening on each axis; cabinet and cavalier obliques keep the frontal face true and recede at 45° (cabinet halves the receding axis).

In one-point perspective the front face is parallel to the picture plane and parallel lines recede to a single vanishing point; in two-point perspective two sets of horizontals vanish to two points on the horizon.

Vanishing point of a family of parallel lines is the intersection of the picture plane with the line through the station point parallel to them.

Cone of vision: the picture should subtend about 30°–60° at the eye (60° maximum) to avoid distortion; station point distance controls this.

Measuring points on the horizon allow true lengths to be transferred in perspective; each principal direction has its own measuring point.

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 shadow of a vertical edge is found by joining the base and top shadows.

Visual weight and balance: symmetrical composition distributes equal mass about an axis; asymmetrical balance compensates a large element with a smaller, denser or higher-contrast one further from the centre.

Rule of thirds divides the frame into a 3×3 grid; placing key elements on the intersections produces dynamic tension compared with central placement.

Colour systems: additive RGB (light) mixes to white; subtractive CMY (pigment) mixes to black; HSV/HSL describe hue angle, saturation and lightness independently of device.

Complementary colours sit opposite on the colour wheel and maximise contrast; analogous colours sit adjacent and produce harmony.

Simultaneous contrast: a colour appears shifted toward the complement of its surroundings, so identical swatches read differently on different grounds.

Golden section and root rectangles (√2, √3, √5) are proportion systems used to lay out formats; the √2 rectangle is the basis of ISO paper sizes.

Gestalt grouping laws - proximity, similarity, continuity, closure, figure-ground - explain how separate marks are read as a single form.

Typographic and layout grid: baseline grid controls vertical rhythm, column grid controls horizontal structure; modular grids subdivide columns into consistent modules.