PAU Technical Drawing II (Castilla y León) cheat sheet

Triangle congruence criteria: SSS, SAS, ASA, and AAS. Two triangles are congruent only when one of these sets is fully determined; SSA does not guarantee congruence (ambiguous case).

In any triangle, the three medians meet at the centroid G, dividing each median in ratio 2:1 from vertex to midpoint. The circumcentre O is equidistant from all three vertices; the incentre I is equidistant from all three sides.

Euler line: circumcentre O, centroid G and orthocentre H are collinear, with OG:GH = 1:2. This holds for every non-equilateral triangle.

Sum of interior angles of a polygon with n sides: (n − 2) × 180°. Sum of exterior angles is always 360°, so each exterior angle of a regular n-gon measures 360°/n.

Regular polygon construction: to inscribe an n-gon in a circle, divide the diameter into n equal parts, then use the standard diagonal (Thales) construction to locate the two vertices on the diameter; the remaining vertices come from arc division.

Diagonals of a quadrilateral: a parallelogram's diagonals bisect each other; a rectangle's diagonals are equal; a rhombus's diagonals are perpendicular; a square satisfies all three.

A quadrilateral is cyclic (inscribable in a circle) if and only if opposite angles sum to 180°. It is tangential (circumscribable) if and only if sums of opposite side lengths are equal: a + c = b + d.

Thales' theorem: an angle inscribed in a semicircle is 90°. Conversely, the locus of points seeing segment AB under a right angle is the circle with diameter AB.

Construction of a regular pentagon: the golden ratio φ = (1 + √5)/2 ≈ 1.618 relates the diagonal d to the side a by d = a·φ. The pentagram's internal intersections divide each diagonal in golden ratio.

Circle tangent to two lines: its centre lies on an angle bisector of the two lines. For two circles, the centre of a tangent circle lies on a hyperbola (external/internal tangency) or on a circle (equal-radius case).

Tangent to a circle from an external point P: the tangent point lies on the circle with diameter OP (O = circle centre). This is Thales' theorem applied to tangency.

Common tangents of two circles: two external and two internal tangents exist when circles are separate; only two external when externally tangent; none when one circle is inside the other.

Locus of points at a fixed distance d from a given circle (centre O, radius r): two concentric circles of radii r + d and |r − d|.

Locus of points equidistant from two points A and B: perpendicular bisector of AB. Locus equidistant from two lines: the pair of angle bisectors.

Conic as locus: ellipse = points with sum of distances to two foci constant (2a); hyperbola = absolute difference of distances constant (2a); parabola = equal distance to focus and directrix.

Linkages: a four-bar linkage is a Grashof crank-rocker when the shortest link plus the longest is ≤ the sum of the other two; then at least one link can fully rotate.

Construction of a tangent circle of given radius r to a line and a circle: offset the line by r and the circle by r (as in the locus above); intersection of the two offset loci gives the tangent circle's centre.

Ellipse: x²/a² + y²/b² = 1, with c² = a² − b². Foci at (±c, 0); eccentricity e = c/a < 1; major axis 2a, minor axis 2b.

Parabola: y² = 2px, focus at (p/2, 0), directrix x = −p/2. Eccentricity e = 1; every point is equidistant from focus and directrix.

Hyperbola: x²/a² − y²/b² = 1, with c² = a² + b². Foci at (±c, 0); eccentricity e = c/a > 1; asymptotes y = ±(b/a)x.

Tangent to an ellipse at a point P: the tangent bisects the exterior angle between the focal radii FP and F'P; equivalently, focal radii make equal angles with the tangent.

Tangent to a parabola at P: reflection of the focal radius about the tangent is parallel to the axis. This is the optical property used in parabolic reflectors.

Tangent to a hyperbola at P: the tangent bisects the angle between the focal radii (internal bisector), unlike the ellipse where it bisects the exterior angle.

Construction of an ellipse by the trammel method: a segment of fixed length slides with its ends on two perpendicular axes; a marked point on the segment traces the ellipse.

Construction of a parabola given focus F and directrix d: for each point on d, draw a perpendicular; the intersection with a circle centred on that perpendicular through F gives a parabola point.

Circular directrix of the hyperbola: the circle centred at a focus with radius 2a is the locus of reflections of the other focus across all tangents.

Dihedral system: two perpendicular planes, horizontal (PH) and vertical (PV), intersect at the ground line (LT). Projections: horizontal projection on PH, vertical projection on PV.

A point's coordinates are given by (x, y, z): x = distance from the vertical plane (depth), y = distance from the horizontal plane (height), z = distance from the profile plane.

A line is true length in a projection when it is parallel to that projection plane. If parallel to LT, it appears true length in both PH and PV.

A line perpendicular to one projection plane projects as a point on that plane (projecting line). Perpendicular to PH: appears as a point in PH; true length in PV.

Plane representation: by its traces (intersection lines with PH and PV). A plane's horizontal trace and vertical trace meet at LT when the plane is perpendicular to the profile plane.

A line is parallel to a plane if it is parallel to some line in the plane. A line is perpendicular to a plane if it is perpendicular to two non-parallel lines of the plane.

Intersection of a line and a plane: pass an auxiliary projecting plane through the line, find its intersection line with the given plane, then intersect that line with the given line.

Two planes intersect in a line; the intersection line's projections pass through the intersections of the trace pairs (with auxiliary planes if the traces are parallel).

Distance from a point to a plane: through the point, draw the perpendicular to the plane; the true length of the segment between the point and the plane is found by the right-triangle method (difference of coordinates).

Regular polyhedra: tetrahedron (4 triangles), cube (6 squares), octahedron (8 triangles), dodecahedron (12 pentagons), icosahedron (20 triangles). Euler: V − E + F = 2.

Section of a polyhedron by a plane is a polygon; its vertices are the intersections of the section plane with the solid's edges. Each vertex is found individually.

True shape of a section: rotate the section plane about its trace until parallel to a projection plane (rabatment), or use change of projection plane, then draw the section in true size.

Development of a prism: unfold lateral faces as rectangles whose widths equal the true edge lengths of the base polygon and whose height is the prism's height.

Development of a cylinder: rectangle of width 2πr and height h, plus two circles of radius r. For an oblique cut, mark ordinates along the development from the true-shape section.

Development of a cone: sector of radius = slant height g and angle α = 360°·r/g, where r is base radius. For a truncated cone, subtract the apex's small sector.

Section of a sphere by a plane is a circle; its true radius is √(R² − d²), where d is the distance from the sphere's centre to the plane.

Right-triangle (difference of coordinates) method: a segment's true length is the hypotenuse of a right triangle whose legs are the segment's projection length and the difference of the endpoints' distances to the projection plane.

Isometric projection: three axes at 120°; equal foreshortening along all three. In true isometric drawing, edges are drawn at full scale (natural scale, factor 1), which is technically an enlargement of 1.22 relative to true isometric projection.

Cavalier perspective (caballera): frontal face drawn true shape; receding axis usually at 45° with reduction factor 1 (cavalier) or 1/2 (cabinet).

Cabinet perspective: receding axis at 45° with foreshortening 1/2; frontal plane in true shape.

Dimetric and trimetric projections: two or three different foreshortening factors respectively; isometric is the special case with three equal factors.

Conical (central) perspective: elements are the station point (SP), picture plane (PP), horizon line (HL), ground line (GT) and vanishing points (VP). Parallel lines in space meet at a common vanishing point.

In one-point perspective, lines perpendicular to the picture plane converge at the principal point; in two-point perspective, two families of horizontals converge at two vanishing points on the horizon.

Scale in perspective: the height of an object on the picture plane is proportional to its distance from the station point; objects farther from the viewer appear smaller.

ISO 5455 standard scales: enlargement 2:1, 5:1, 10:1; reduction 1:2, 1:5, 1:10, 1:20, 1:50, 1:100, 1:200, 1:500, 1:1000. Scale = drawing dimension / real dimension.

Dimension lines: thin continuous line with arrowheads at both ends; the dimension value is written above the line, readable from the bottom or right of the sheet.

Line types (ISO 128): continuous thick = visible edges; continuous thin = dimension, extension, hatching lines; dashed = hidden edges; dash-dot thin = centre lines and axes; continuous thin with zigzag = long break lines.

Hatching (section lines): thin continuous lines at 45° to the main outline, equally spaced. Adjacent parts in an assembly are hatched at different angles or spacings.

Dimensioning rules: never repeat a dimension; do not dimension to hidden lines if avoidable; place dimensions outside the view where possible; use one datum for related dimensions.

Section views: full section (cutting plane through the whole object), half section (half view in section, half outside), offset section (cutting plane bends to pass through features), broken-out section (partial, bounded by a freehand or irregular break line).

Tolerances: ISO 286 fits use a letter (hole: uppercase; shaft: lowercase) and an IT grade number, e.g. H7/g6. Clearance fit: hole minimum ≥ shaft maximum. Interference fit: shaft minimum > hole maximum.

Metric thread designation: M followed by nominal diameter and pitch, e.g. M10×1.5. Coarse pitch is omitted (M10 means coarse). A left-hand thread is marked LH.

Thread representation: external thread - thick continuous line for the crest, thin continuous line for the root, both at the nominal diameter and root diameter respectively; the thread run-out is shown by a thick line.

Surface roughness (ISO 4287/1302): Ra is the arithmetic mean deviation of the profile. The symbol is a check mark with the value written above it; a circle in the hook means all surfaces, a bar means machined.

Geometric tolerances (ISO 1101): symbols for straightness (-), flatness (▱), circularity (○), cylindricity (⌭), perpendicularity (⊥), parallelism (∥), position (⌖), concentricity (◎).

Bolted joint representation: bolt head and nut drawn with simplified proportions - nut height ≈ 0.8d, bolt head height ≈ 0.7d, washer thickness ≈ 0.15d.

Screw thread simplified representation: in the longitudinal view, the crest is a thick line and the root a thin line; in the end view, the root is a thin circle open about 3/4 of the circumference.

Tolerance grades IT01 to IT18: smaller numbers mean tighter tolerance. IT grades are determined by nominal size and grade; fundamental deviations are given by letters A–ZC.

Freehand sketching proportions: use a light construction box and centre lines first, then darken visible outlines; keep line weight hierarchy (thick visible, thin construction).

CAD layers: separate geometry (visible, hidden, centre, dimensions, hatching) onto distinct layers with distinct line types and colours; ISO 128 line types should be respected on the plot.

CAD coordinate entry: absolute (x,y), relative (@dx,dy), polar (@length<angle). Object snaps (endpoint, midpoint, intersection, perpendicular, tangent) enforce geometric accuracy.

Orthographic projection conventions: first-angle (European, ISO) places the top view below the front view; third-angle (American, ASME) places it above.

Isometric sketching on isometric grid: verticals stay vertical, the other two axes at 30° above horizontal; circles appear as ellipses with major axis 1.22d and minor 0.71d.

Visual analysis of a drawing: check for missing or duplicated dimensions, mismatched projections between views, incorrect line types, and inconsistent scale before reading geometry.

CAD blocks and standard parts: reusable symbols (fasteners, bearings, threads) stored as blocks with attributes; libraries must follow ISO or the applicable national standard.

The centroid divides each median 2:1 from vertex to midpoint; it is the centre of gravity of the triangle's area.

The orthocentre is the intersection of the three altitudes; in an obtuse triangle it lies outside the triangle.

The incentre is the intersection of the internal angle bisectors; its distance to each side is the inradius r = A/s, where A is area and s is semiperimeter.

The circumcentre is the intersection of the perpendicular bisectors; its distance to each vertex is the circumradius R = abc/(4A).

Euler's relation: OI² = R² − 2Rr, linking circumcentre O, incentre I, circumradius R and inradius r.

Two circles are externally tangent when the distance between centres equals the sum of radii; internally tangent when it equals the difference.

A line tangent to a circle is perpendicular to the radius at the point of tangency.

The radical axis of two circles is the locus of points with equal power (tangent length) to both circles; it is perpendicular to the line of centres.

Inversion: a circle through the inversion centre maps to a line; tangency is preserved under inversion, which is a standard tool for Apollonius problems.

Apollonius problem: circle tangent to three given objects (points, lines, circles) - up to 8 solutions, found by reducing to line–circle cases via inversion or homothety.

Ellipse from conjugate diameters: the parallelogram method or the concentric-circle (Circles of Apollonius) method gives points and tangents.

Parabola from focus and directrix: pick points on the directrix, draw perpendiculars, intersect with circles centred on the directrix through F.

Hyperbola from foci and transverse axis: for each point, the difference of distances to the foci equals 2a.

Tangent to a conic from an external point: use the polar line of the point with respect to the conic; the tangent points are the intersections of the conic with the polar.

A conic is determined by five points (or four points and one tangent); this is the basis of the Pascal/Brianchon constructions.

Fundamental deviation letters: A–H are holes below the zero line; J–ZC holes above; a–h shafts above the zero line; j–zc shafts below.

H7/g6 is a common clearance fit for sliding parts; H7/p6 is a typical interference fit for press fits.

Ra values: 0.4, 0.8, 1.6, 3.2, 6.3, 12.5 µm are common preferred values; lower Ra means smoother surface.

Datum features are indicated by a capital letter in a square box connected to a filled or open triangle on the surface.

A tolerance frame reads left to right: geometric symbol, tolerance value, datum letter(s).