Canary Islands PAU Technical Drawing II cheat sheet

Triangle construction from three sides (a, b, c): draw side a = BC; with centre B radius c and centre C radius b, the arc intersection is A. Existence requires a < b + c, a > |b − c|.

Circumcentre O: intersection of the three perpendicular bisectors; equidistant from all vertices, so it is the centre of the circumcircle. It lies inside an acute triangle, on the hypotenuse midpoint for a right triangle, outside for obtuse.

Incentre I: intersection of the angle bisectors; equidistant from the three sides, so it is the centre of the incircle. For a triangle with sides a, b, c and semiperimeter s, inradius r = Area / s.

Orthocentre H: intersection of the three altitudes. Euler line: O, centroid G and H are collinear with OG : GH = 1 : 2, and OH² = 9R² − (a² + b² + c²).

Centroid G: intersection of the medians, each divided in ratio 2 : 1 from the vertex. Coordinates are the mean of the three vertex coordinates; the medial triangle has one quarter the area of the original.

Regular n-gon: central angle = 360°/n, interior angle = 180° − 360°/n, exterior angle = 360°/n. Side from circumradius R is s = 2R·sin(180°/n); apothem a = R·cos(180°/n).

Square on a given side: erect perpendiculars at both endpoints, mark the side length on each, join. Square on a given diagonal d: its side is d/√2; construct the perpendicular bisector of d and lay off d/2 each way from the midpoint.

Rectangle or parallelogram from two sides and the included angle: lay off both sides from the vertex, then transfer each length with the compass from the opposite endpoint to close the figure.

Construct a regular polygon on a given side by dividing a semicircle on that side into n equal parts and drawing the ray through the second division point to locate the circumcentre.

Translation by vector v: P′ = P + v. It is an isometry preserving orientation; lines map to parallel lines and the fixed points are none (identity only when v = 0).

Rotation about centre O through angle θ: OP′ = OP and angle POP′ = θ. It is an isometry preserving orientation; the only fixed point is O, and composing two rotations gives a rotation through the sum of the angles (or a translation if the sum is 360°).

Axial symmetry (reflection in a line): reverses orientation; the axis is pointwise fixed. Composition of two reflections in parallel lines is a translation of twice the separation; in intersecting lines it is a rotation of twice the angle between them.

Central symmetry (point reflection): equivalent to a rotation of 180° about the centre; reverses orientation, and the centre is the midpoint of every segment PP′.

Homothety centre O, ratio k: OP′ = k·OP with P′ on line OP. It preserves angles and orientation, maps lengths by |k| and areas by k²; k < 0 gives an inverse homothety on the opposite ray.

Similarity (scale + isometry): corresponding angles equal, corresponding lengths in ratio k, areas in ratio k², volumes in ratio k³. Two triangles are similar with AA, SSS in ratio, or SAS in ratio.

Inversion centre O, power r²: O, P, P′ are collinear and OP·OP′ = r². It is conformal (angle-preserving) but not an isometry; lines through O map to themselves, other lines map to circles through O, and circles through O map to lines.

Fixed points of an inversion are the points of the circle of inversion; points inside map outside and vice versa, and OP → 0 gives P′ → infinity.

Composition of two homotheties with the same centre is a homothety with ratio k₁·k₂; with distinct centres it is a homothety whose centre lies on the line joining them (or a translation when k₁·k₂ = 1).

Tangent from an external point P to a circle of centre O: draw the circle with diameter OP; its intersections T with the given circle are the tangent points, since angle OTP = 90°.

Line tangent to a circle at T is perpendicular to radius OT. Two circles are tangent externally when d = r₁ + r₂ and internally when d = |r₁ − r₂|; the tangent point lies on the line of centres.

Common tangents: two non-overlapping circles have 4 (2 external, 2 internal); externally tangent circles have 3; overlapping circles have 2; internally tangent have 1; one inside the other have 0.

Ellipse: set of points with PF₁ + PF₂ = 2a. With semi-axes a, b and focal distance c, a² = b² + c², eccentricity e = c/a < 1, and the focal radii of a point satisfy PF₁ = a + e·x (x measured from the centre).

Parabola: points equidistant from focus F and directrix. With vertex at origin and axis along x, y² = 2px, where p is the focal parameter; a point's distance to the focus equals its distance to the directrix.

Hyperbola: |PF₁ − PF₂| = 2a, with c² = a² + b², e = c/a > 1, and asymptotes y = ±(b/a)x. The rectangle 2a × 2b centred at the origin gives the asymptotes through its diagonals.

Tangent to a conic at a point on it: bisects the angle between the focal radii for the ellipse (external bisector) and hyperbola, and for the parabola the tangent bisects the angle between the focal radius and the line parallel to the axis.

Circular-directrix construction: the points of a conic are the centres of circles through the focus that are tangent to the directrix circle (or line); this generates the curve point by point with compass and rule.

Tangent to a conic from an external point: draw the circle with diameter P–F, intersect with the focal circle (centre F, radius 2a) for the ellipse/hyperbola; the lines from P to those intersection points are the tangents.

A point is given by two projections; a third projection is obtained by transferring the distance to the reference line consistently (same distance from the ground line in both views for the depth coordinate).

A line is fully determined by its two projections. True length appears only when the line is parallel to a projection plane; otherwise use the right-triangle method: true length² = (projected length)² + (difference of the missing coordinates)².

A point lies on a line when its projections lie on the corresponding projections of the line. A line lies in a plane when two of its points lie in the plane, or when its traces lie on the plane's traces.

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

Intersection of two planes: find where two pairs of traces meet, or cut both with two auxiliary parallel planes and join the two resulting intersection points.

Parallelism: a line is parallel to a plane if it is parallel to some line in the plane; two planes are parallel if two intersecting lines of one are parallel to two intersecting lines of the other.

Perpendicularity: a line is perpendicular to a plane when its projections are perpendicular to the corresponding traces (or to the principal lines) of the plane; a plane is perpendicular to another when it contains a line perpendicular to it.

Distance from a point to a plane: drop the perpendicular, find the piercing point by the auxiliary-plane method, then obtain the true length of the segment by the right-triangle method.

Distance between two skew lines: construct the common perpendicular through an auxiliary plane parallel to one line containing the other; its true length is the shortest distance.

Section of a polyhedron by a plane: intersect the plane with each face plane (or each edge) and join the points in face order; the section of a convex polyhedron by a plane is a convex polygon.

True shape of a section: choose a new projection plane parallel to the cutting plane (change of plane), then project the section orthogonally onto it; all its segments appear at true length.

Development of a prism: unfold the lateral faces as consecutive rectangles whose widths are the true lengths of the base edges, obtained from a true-shape view of the base; add the two bases.

Development of a pyramid or cone: lay out the true lengths of the lateral edges (or generators) as radii; for a right cone the sector angle is 360°·r/g, where r is base radius and g the slant height.

Development of a cylinder: a rectangle of width 2πr and height h, plus two circles; a point on the surface is located by its generating line and its height.

Intersection of a polyhedron with a line: use an auxiliary projecting plane through the line; the entry and exit points are where the line crosses the section polygon of the solid by that plane.

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

Visibility rule: in each view an edge is hidden where it lies behind a face; test by comparing the depth coordinate of the edge and the face at their apparent crossing point.

First-angle (European) projection: the object lies between observer and plane, so the view is projected onto the far side. The top view is placed below the front view, and the left-side view is placed on the right.

Third-angle (American) projection: the projection plane lies between observer and object. The top view is placed above the front view, and the right-side view is placed on the right; the symbol shows a truncated cone with the small circle nearest the front view.

Alignment: front and top views share the same width (vertical alignment), front and side views share the same height, and top and side views share the same depth, transferred with a 45° mitre line.

Auxiliary view: project onto a plane perpendicular to one principal plane and inclined to another; it shows the true shape of an inclined face. An auxiliary view taken perpendicular to a line gives that line's true length.

Dimensioning rules: each dimension appears once; place it on the view showing the feature's true shape; avoid dimensions on hidden lines; use diameter Ø for full circles and radius R for arcs, with the leader through or towards the centre.

Scale: scale = drawing length / real length. 1:50 means 1 unit on paper represents 50 units in reality; a 2 m object at 1:50 measures 40 mm. Enlargement scales are written 2:1, 5:1.

Line hierarchy: continuous thick for visible edges and outlines, continuous thin for dimension, extension and hatching lines, dashed for hidden edges, chain thin for centre lines and axes, and a freehand thin line for break lines.

Reference dimensions are given once from a datum; chain dimensioning accumulates error, so datum (parallel) dimensioning is preferred where tolerance stack-up matters.

Isometric projection: three axes at 120° to each other, with equal foreshortening. In true isometric drawing the axes are drawn at 30° to the horizontal and lengths are drawn full size (isometric drawing, not projection).

Isometric projection foreshortens all three axes by cos 35°16′ ≈ 0.816; isometric drawing uses the full length, so it is 1/0.816 ≈ 1.22 times larger than the true projection.

Dimetric projection: two axes equally foreshortened, the third different. Trimetric: all three foreshortening factors differ. Both are specified by the angles of the axes.

Cavalier perspective (oblique): the front face is drawn true shape in the picture plane; receding edges are drawn at a chosen angle (commonly 45°) at full scale. Cabinet perspective uses one half scale on the receding axis.

In oblique projection the receding axis is usually drawn at 30°, 45° or 60° to the horizontal; 45° with half scale (cabinet) reduces the apparent distortion of depth.

Circles in isometric view become ellipses: on each face the major axis is perpendicular to the axis normal to that face and equals 1.22·d (isometric drawing), the minor axis equals 0.71·d.

Isometric ellipse construction: enclose the circle in a square, draw the isometric square (rhombus), and use the four-centre method with centres at the obtuse corners and on the long diagonal.

Hidden lines are normally omitted in axonometric and oblique pictorials; axes are drawn thin and the object outlines thick, with hatching or shading used only to indicate separate parts.

ISO A-series formats: A0 = 1 m² with sides in ratio 1:√2 (841 × 1189 mm); each subsequent size halves the longer side (A1 594 × 841, A2 420 × 594, A3 297 × 420, A4 210 × 297).

Preferred scales: 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. The scale is stated in the title block.

Line types per ISO 128: continuous thick (visible outlines), continuous thin (dimension, extension, hatching, leader lines), dashed (hidden outlines), chain thin (centre lines, axes, cutting-plane traces), continuous wavy (limits of partial views).

Dimension line terminators: filled arrowheads are standard; oblique strokes or dots are permitted alternatives. The dimension line is drawn parallel to the measured feature, with extension lines starting about 1 mm from the object.

Numbers: dimension figures are placed above the dimension line, readable from the bottom or from the right-hand edge of the sheet; the unit (mm) is stated once in the title block and omitted from individual dimensions.

Tolerances and fits: ISO system of limits and fits uses hole basis (H) or shaft basis (h); a fit such as H7/g6 gives the tolerance class and letter for hole and shaft, with uppercase for holes and lowercase for shafts.

Geometrical tolerancing (ISO 1101) uses a rectangular frame: symbol, tolerance value, and datum letters. Symbols include straightness, flatness, circularity, cylindricity, parallelism, perpendicularity, position and concentricity.

Surface texture is indicated by the ISO 1302 check mark with the Ra value in micrometres; the general note such as Ra 3.2 applies where no individual symbol is shown.

Title block content: drawing number, title, scale, projection method symbol, sheet size, material, mass, tolerances, and approval signatures; it is normally placed at the bottom right of the sheet.