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

Thales' theorem: parallel lines cutting two transversals divide them into proportional segments, so AB/BC = A'B'/B'C'. Converse: equal ratios on two lines imply the joining segments are parallel.

Fourth proportional to a, b, c: construct with two rays from a vertex, mark a and b on one ray, c on the other, then draw the parallel giving x = b·c/a.

Third proportional to a and b satisfies a/b = b/x, so x = b²/a; the mean proportional (geometric mean) m satisfies a/m = m/b, so m = √(ab).

Golden section of segment AB: with M the midpoint, MB perpendicular of length AB/2, circle centered M through B cuts AM at N; then AN/AB = 1/φ with φ ≈ 1.618.

Scales of similarity: similar figures with ratio k have areas in ratio k² and volumes in ratio k³; corresponding angles are equal and corresponding sides proportional.

Homothety (center O, ratio k): every point P maps to P' on line OP with OP' = k·OP; lines map to parallel lines, and k < 0 gives an inverted figure through O.

Inversion (center O, power k): P and P' are collinear with O and OP·OP' = k; a circle not through O inverts to a circle, and a line not through O inverts to a circle through O.

Rotation by angle α about center O: use perpendicular bisectors of AA' to locate O when α is known, or bisect angle AOA' to find α when O is known.

Translation is defined by a vector; a glide reflection combines a reflection in an axis with a translation parallel to that axis and has no fixed point.

Tangency condition: two circles are tangent when the distance between centers equals the sum of radii (external) or the absolute difference (internal); the point of tangency lies on the line of centers.

Circle tangent to two given lines: its center lies on an angle bisector of the lines; add a radius offset to a parallel line to fix the center for a given radius.

Circle of radius r tangent to a given circle and passing through point P: center lies on a circle centered at P of radius r and on a concentric circle of radius R ± r from the given circle's center.

Ellipse: sum of focal distances is constant 2a; with semiaxes a, b the focal distance is c = √(a² − b²). Tangent at a point bisects the angle between the focal radii; the tangent is perpendicular to the bisector of the angle between the point and the two foci.

Hyperbola: absolute difference of focal distances is constant 2a, with c = √(a² + b²); the asymptotes pass through the center with slopes ±b/a.

Parabola: points are equidistant from focus F and directrix d; the tangent at a point bisects the angle between the focal radius and the line parallel to the axis.

Osculating circle (circle of curvature) at a point of a curve: center lies on the normal; for a parabola at the vertex its radius equals the focal parameter p.

Cycloid: generated by a point on a circle rolling on a line; one arch has length 8r and area 3πr², and the tangent at a point passes through the top of the generating circle.

Involute of a circle: generated by unwinding a taut string from the circle; the tangent to the involute is perpendicular to the string, hence tangent to the base circle.

A point is fixed by two projections; a point in the second quadrant has both projections above the ground line, and the horizontal and vertical projections always lie on a common perpendicular to the ground line.

A line is determined by its two projections; its traces are the points where it meets the horizontal and vertical planes, and both traces lie on the same projection line.

A plane is given by its traces (intersections with the horizontal and vertical planes); traces meet on the ground line unless the plane is parallel to one projection plane.

Intersection of two planes: find two points each lying in both planes by intersecting corresponding traces, then join them; if traces are parallel the intersection line is parallel to them.

Intersection of a line with a plane: pass an auxiliary plane through the line, intersect it with the given plane, and the meeting of that intersection line with the given line is the piercing point.

A line is parallel to a plane when it is parallel to some line of that plane; a line perpendicular to a plane has its projections perpendicular to the corresponding traces of the plane.

Visibility in dihedral: a projection is hidden where the point lies behind a solid or plane; determine it by comparing depths (horizontal projection) or heights (vertical projection).

Change of projection plane: the new ground line is drawn perpendicular to the direction that must become true length; distances to the old plane are transferred unchanged to the new one.

A plane perpendicular to both projection planes appears as a line in both projections; a plane parallel to the ground line has parallel traces.

True length of a segment: use the difference of depths and the projected length as legs of a right triangle, or rotate/change planes until the segment is parallel to a projection plane.

True size of a plane figure: change of plane or rotation until the plane is parallel to the projection plane; two successive changes are usually needed for an oblique plane.

Distance from a point to a plane: drop the perpendicular from the point to the plane, find its piercing point, then obtain the true length of that perpendicular segment.

Distance between two parallel planes or a plane and a parallel line equals the length of any common perpendicular, measured in true magnitude.

Distance between two skew lines: construct the common perpendicular by passing a plane through one line parallel to the other and dropping a perpendicular from a point of the second line.

Theorem of the three perpendiculars: if a line in a plane is perpendicular to the projection of an oblique line, it is also perpendicular to the oblique line itself.

A plane perpendicular to a line contains every line perpendicular to that line; its traces are perpendicular to the corresponding projections of the line.

Angle between a line and a plane: project the line onto the plane and measure the angle between the line and its projection in true magnitude.

Dihedral angle between two planes: cut both with a plane perpendicular to their intersection line and measure the plane angle obtained.

Euler's formula for convex polyhedra: V − E + F = 2, where V, E, F are vertices, edges, and faces; the five regular polyhedra are tetrahedron, cube, octahedron, dodecahedron, and icosahedron.

Regular tetrahedron of edge a: height a√6/3, total surface a²√3, volume a³√2/12; the cube has 12 edges, 8 vertices, 6 faces, and body diagonal a√3.

Regular octahedron of edge a: volume a³√2/3; regular dodecahedron and icosahedron are duals, and their dihedral angles are approximately 116.57° and 138.19°.

Prism and cylinder: lateral surface equals perimeter of the base times height; volume equals base area times height; a right section gives the development of an oblique prism.

Pyramid and cone: volume equals one third of base area times height; a truncated cone's lateral surface equals π(R + r)·g, where g is the slant height.

Sphere: surface 4πR², volume 4πR³/3; a plane section is a circle of radius √(R² − d²), where d is the distance from the center to the plane.

Cone development: lateral surface is a circular sector of radius equal to the generatrix and angle α = 360°·r/g, where r is the base radius and g the generatrix.

Torus: generated by a circle of radius r whose center is at distance R from the axis; volume 2π²Rr² and surface 4π²Rr.

Intersection of a sphere with a cylinder or cone: use auxiliary planes parallel to the axis (cylinder) or containing the axis (cone) to obtain points of the curve.

Axonometric projection is a parallel projection onto a plane not parallel to the coordinate axes; the three axes are drawn with a common origin and scale coefficients along each axis.

Isometric perspective: the three axes form 120° angles on the drawing, all scale coefficients equal 0.816 in true isometric projection, rounded to 1 in common isometric drawing.

Dimetric and trimetric systems use two or three different scale coefficients; the axonometric axes are obtained by rotating the coordinate trihedron and projecting it.

Cavalier perspective (caballera): the frontal plane is drawn true size, and the receding axis is drawn at a chosen angle (commonly 45° or 30°) with a reduction coefficient, often 0.5.

Caballera reduction coefficient: for a 45° receding axis the natural coefficient is 0.5; for 30° it is about 0.816, and for 60° about 0.577, applied to depths only.

Isometric ellipse: a circle in a coordinate plane projects as an ellipse with major axis 1.22·d and minor axis 0.71·d for diameter d; the major axis is perpendicular to the third axonometric axis.

Hidden-line removal in axonometric views uses the same reasoning as dihedral: determine which edges lie behind faces, and dash them.

Axonometric scale: a segment parallel to an axis is drawn multiplied by that axis's coefficient, so true lengths are recovered by dividing by the coefficient.

ISO A-series paper: A0 has an area of 1 m² and sides in ratio √2; each subsequent size halves the larger side, so A4 is 210 × 297 mm.

Line types: continuous thick for visible edges, dashed for hidden edges, chain line for axes and symmetry, continuous thin for dimension and extension lines, and freehand for break lines.

Scale is drawing length divided by real length; common scales are 1:1, 1:2, 1:5, 1:10, 1:20, 1:50, and enlargements 2:1, 5:1, 10:1.

Dimensioning rules: dimensions in true size regardless of scale, placed above a dimension line parallel to the measured edge, with arrowheads at both ends and extension lines starting 1–2 mm from the object.

Units are stated once in the drawing frame (millimetres by default) and omitted from individual dimensions; diameters take the symbol Ø, radii R, and spheres SØ or SR.

Angular dimensions use an arc dimension line centered on the vertex; counts and repetitions use the form 4×Ø10, and chamfers are noted as 1×45°.

Projection symbols: first-angle (European) places the left view on the right side and is marked by a truncated cone symbol; third-angle (American) reverses the arrangement.

Sketching: proportions are checked by holding a pencil at arm's length and comparing apparent lengths, and construction lines remain light while final outlines are reinforced.

Sections and cuts: hatch cut surfaces with thin parallel lines at 45°, omit hatching on shafts, bolts, and ribs cut longitudinally, and label the cutting plane with arrows showing view direction.

Representation of a polyhedron in dihedral: give the projections of all vertices, join edges respecting visibility, and check that each face is planar by verifying its vertices are coplanar.

Regular pyramid on a horizontal base: the base is true size in the horizontal projection, the apex lies on a vertical through the base center, and edges appear as lines to that point.

Cylinder in dihedral: the horizontal projection of a vertical cylinder is a circle and the vertical projection is a rectangle whose width is the diameter; the contour generatrices are the extreme ones.

Cone in dihedral: the base is a circle in one projection and the apex is projected on its center; the vertical projection is a triangle bounded by extreme generatrices.

Sphere in dihedral: both projections are circles of the true radius, but the visible contour is not a meridian; the equator and a principal meridian are drawn to fix the surface.

Section of a polyhedron by a plane: intersect the plane with each face, giving a polygon whose vertices lie on the edges; obtain its true size by change of plane.

Section of a cylinder or cone by an oblique plane gives an ellipse, parabola, or hyperbola depending on whether the plane cuts all generatrices, is parallel to one, or cuts both nappes.

Development of a polyhedron: unfold faces about chosen edges keeping true sizes; edges shared by two faces must have equal length in both unfolded copies.

Points on a surface of revolution lie on parallels (circles perpendicular to the axis); a point is located by giving its parallel and its position along a generatrix.

ISO A-series paper: A0 has an area of 1 m² and sides in ratio √2; each subsequent size halves the larger side, so A4 is 210 × 297 mm.

Line types: continuous thick for visible edges, dashed for hidden edges, chain line for axes and symmetry, continuous thin for dimension and extension lines, and freehand for break lines.

Scale is drawing length divided by real length; common scales are 1:1, 1:2, 1:5, 1:10, 1:20, 1:50, and enlargements 2:1, 5:1, 10:1.

Dimensioning rules: dimensions in true size regardless of scale, placed above a dimension line parallel to the measured edge, with arrowheads at both ends and extension lines starting 1–2 mm from the object.

Units are stated once in the drawing frame (millimetres by default) and omitted from individual dimensions; diameters take the symbol Ø, radii R, and spheres SØ or SR.

Angular dimensions use an arc dimension line centered on the vertex; counts and repetitions use the form 4×Ø10, and chamfers are noted as 1×45°.

Projection symbols: first-angle (European) places the left view on the right side and is marked by a truncated cone symbol; third-angle (American) reverses the arrangement.

Sketching: proportions are checked by holding a pencil at arm's length and comparing apparent lengths, and construction lines remain light while final outlines are reinforced.

Sections and cuts: hatch cut surfaces with thin parallel lines at 45°, omit hatching on shafts, bolts, and ribs cut longitudinally, and label the cutting plane with arrows showing view direction.