Catalonia PAU Technical Drawing (Dibuix Tècnic II) cheat sheet

Thales' theorem: an angle inscribed in a semicircle is a right angle; the midpoint of the hypotenuse is equidistant from all three vertices.

Tangent from an external point P to a circle of center O: draw the circle with diameter OP; its intersections with the given circle are the tangency points.

Common external tangent to two circles: construct a circle centered at the larger circle's center with radius R−r; tangents from the smaller center to this auxiliary circle give the tangent directions.

Ellipse construction: with foci F1, F2 and major axis 2a, every point P satisfies PF1 + PF2 = 2a; the minor semiaxis b satisfies b² = a² − c².

Parabola: points equidistant from focus F and directrix d; the tangent at P bisects the angle between PF and the line through P parallel to the axis.

Hyperbola: |PF1 − PF2| = 2a; the asymptotes pass through the center with slopes ±b/a, where b² = c² − a².

Homothety of center O and ratio k maps a circle to a parallel circle; the ratio of radii equals |k| and the centers are collinear with O.

Inversion of center O and power k maps a circle not through O to another circle; a circle through O maps to a line not through O.

Golden ratio φ = (1+√5)/2 ≈ 1.618; in a regular pentagon the diagonal-to-side ratio equals φ.

A point is represented by its horizontal projection (below the ground line) and vertical projection (above it); the distance to the ground line equals the distance to the corresponding projection plane.

A line's true length appears in a projection only when the line is parallel to that projection plane; otherwise use the right-triangle method with the projection and the difference of distances to the plane.

A plane is defined by two intersecting lines, two parallel lines, a line and an external point, or three non-collinear points; its traces are the lines where it meets the projection planes.

Intersection of two planes: find the piercing points of one plane's trace lines with the other plane; the line through these points is the intersection.

Intersection of a line with a plane: pass an auxiliary projecting plane through the line, intersect it with the given plane, and locate where the line meets that intersection line.

A line perpendicular to a plane has its horizontal projection perpendicular to the plane's horizontal trace, and its vertical projection perpendicular to the vertical trace.

Change of projection plane: project the existing coordinates onto a new ground line; distances to the new plane are transferred perpendicular to the new ground line.

Rotation method: rotate a line about an axis perpendicular to a projection plane until it is parallel to the other plane; the true length appears in the second projection.

Euler's formula for convex polyhedra: V − E + F = 2.

Regular tetrahedron: 4 equilateral triangular faces; height h = a√(2/3) ≈ 0.8165a; volume V = a³/(6√2).

Regular octahedron: 8 equilateral triangular faces; volume V = (√2/3)a³; the distance between opposite vertices is a√2.

Cube: 6 square faces, 12 edges, 8 vertices; body diagonal = a√3; volume = a³.

Cylinder: lateral area = 2πrh; total area = 2πr(h+r); volume = πr²h. Cone: lateral area = πrg (g = slant height); volume = (1/3)πr²h.

Sphere: area = 4πr²; volume = (4/3)πr³; a plane section at distance d from the center is a circle of radius √(r²−d²).

Section of a polyhedron by a plane: find the intersection points of the plane with the edges; connect points lying on the same face; the section is a polygon.

Development of a right cylinder: rectangle of width 2πr and height h; development of a right cone: circular sector of radius g and angle θ = 360°·(r/g).

Truncated cone development: annular sector with radii equal to the slant distances from the apex to the cut circles, and angle θ = 360°·(R−r)/slant height difference.

Dimension lines are drawn parallel to the measured edge, with arrowheads at both ends; extension lines start about 1 mm from the object and extend about 2 mm beyond the dimension line.

Numerals are placed above the dimension line, centered, and read from the bottom or from the right-hand side of the drawing; they are never crossed by lines.

Scale = drawing length / real length; recommended scales include 1:1, 1:2, 1:5, 1:10, 2:1, 5:1; the scale is indicated in the title block.

First-angle projection (ISO-E): the object is between the observer and the plane; the view from the left is placed on the right, and the top view is placed below the front view.

Third-angle projection (ISO-A): the projection plane is between the observer and the object; the top view is placed above the front view and the left view on the left.

A cut is used when internal features are not clear from hidden lines; the cutting plane is indicated by a chain line with arrows showing the viewing direction and labeled with capital letters.

Hatching in a section is drawn with thin continuous lines at 45° to the main outline; adjacent parts are hatched in opposite directions or at different spacings.

A half-section shows one half in section and the other half in outside view, separated by a center line; it is used for symmetrical parts.

Hidden lines are omitted in sectional views unless needed for clarity; center lines are chain lines with long and short dashes.

Isometric projection: the three axes form 120° angles; the foreshortening factor along each axis is √(2/3) ≈ 0.816, but isometric drawing uses full scale along the axes.

Dimetric projection: two axes have equal foreshortening; trimetric projection: all three foreshortening factors differ.

Cabinet oblique projection: the front face is drawn true shape; the receding axis is usually at 45° and is drawn at half scale.

Cavalier oblique projection: the receding axis is drawn at full scale; the front face remains true shape.

In one-point perspective, the front face is parallel to the picture plane; all receding lines converge at a single principal vanishing point.

In two-point perspective, vertical lines remain vertical, and horizontal lines parallel to the two principal directions converge to two vanishing points on the horizon line.

The station point is the observer's eye position; its distance from the picture plane determines the perspective cone angle (typically 30°–60°).

In axonometry, circles in coordinate planes become ellipses; in isometric projection the major axis of each ellipse is 1.22d and the minor axis is 0.71d for a circle of diameter d.

A flowchart uses standard symbols: oval for start/end, rectangle for process, diamond for decision, parallelogram for input/output, and arrows for flow direction.

In a bar chart, the length of the bar is proportional to the value; the baseline must start at zero to avoid misleading comparisons.

In a line graph, the independent variable is placed on the horizontal axis and the dependent variable on the vertical axis; each axis must carry a label and unit.

CAD absolute coordinates are entered as x,y; relative coordinates as @dx,dy; polar coordinates as @distance<angle.

Object snaps (endpoint, midpoint, center, intersection, perpendicular) ensure geometric accuracy; ortho mode constrains cursor movement to horizontal and vertical.

Layers group objects by function or line type; the current layer determines the properties of new objects; line weights are assigned per layer or per object.

Freehand sketching: construct straight lines with light guide lines, then darken the final line; circles are sketched by drawing a square and inscribing arcs.

A Gantt chart represents activities as horizontal bars along a time axis; dependencies are shown with arrows between bars.

A schematic diagram uses standardized symbols and omits physical scale; it communicates function and connections, not dimensions.

A 2H pencil is used for construction and guide lines; an HB or B pencil is used for final visible outlines; a 4H or harder pencil is used for fine construction.

Compass leads are sharpened to a chisel point; the compass is held by the knurled knob and tilted slightly in the direction of travel.

A T-square is used against the drawing board's working edge to draw horizontal lines; a set square is used against the T-square for vertical and 30°/45°/60° lines.

Line hierarchy: visible outlines are thick continuous lines; hidden edges are dashed; axes and center lines are chain lines; dimension and hatching lines are thin continuous lines.

Standard line width ratio is approximately 2:1 between thick and thin lines; a common thick line width is 0.5 mm or 0.7 mm.

Tangency points must be marked and the line must change direction smoothly; a visible gap or kink indicates an inaccurate tangency.

When inking, draw all circles and arcs first, then straight lines; allow ink to dry before erasing construction lines.

To divide a segment into n equal parts, draw an auxiliary ray at an acute angle, mark n equal steps with the compass, and project parallels to the segment's endpoint.