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

Thales' theorem: a line parallel to one side of a triangle cutting the other two sides divides them in proportional segments; AB/BC = A'B'/B'C'.

Mean proportional (geometric mean) of segments a and b: height h of a right triangle satisfies h² = a·b; construct by drawing a semicircle on a+b and erecting a perpendicular at the junction point.

Golden section of segment AB: divide so that AB/AP = AP/PB ≈ 1.618; construction places AP as the side of a square plus the diagonal remainder.

Regular polygon inscribed in a circle: central angle = 360°/n; side length s = 2R·sin(180°/n); apothem a = R·cos(180°/n).

Approximate construction of any regular n-gon: divide a diameter into n equal parts, draw arcs from the diameter endpoints through the second division point; intersections give the vertices.

Tangent to a circle from external point P: the circle with diameter OP (O = centre) intersects the given circle at the two tangency points.

Ellipse as locus: sum of distances to foci is constant (2a); major axis 2a, minor axis 2b, focal distance c with a² = b² + c².

Parabola: points equidistant from focus F and directrix d; vertex is the midpoint between F and d; axis is perpendicular to d through F.

Hyperbola: absolute difference of distances to foci is constant 2a; asymptotes pass through the centre with slope b/a where c² = a² + b².

Dihedral system: horizontal plane H and vertical plane V meet at the ground line; the horizontal projection lies on H, the vertical projection on V, related by a vertical reference line perpendicular to the ground line.

Point coordinates: (x, y, z) where x = distance along the ground line, y = depth (distance to V), z = height (distance to H); in the folded view y appears below the ground line and z above it.

A line is true length in a view when it is parallel to that projection plane; if parallel to H it shows true length in plan and true angle in elevation only when also parallel to V.

A plane appears in true shape in the view perpendicular to its normal; edge-on (degenerate) in the view parallel to its normal.

Change of plane method: draw a new reference line perpendicular to the projection of a line to obtain its true length in one step.

Intersection of two planes: find two points common to both, typically where each plane's traces meet; the line through them is the intersection.

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

Visibility rule: in a view, the element whose depth (or height) coordinate is closer to the observer occludes the other at a crossing.

Isometric projection: three axes at 120°; foreshortening equal on all three; true isometric drawing uses full scale along axes (no 0.816 reduction).

Dimetric and trimetric axonometry differ by the chosen angles between axes; scale factors on each axis follow the chosen projection direction.

Cavalier and cabinet oblique: front face drawn true shape; receding axis at 45° (cabinet uses half scale on the receding axis).

Perspective: objects of equal size subtend smaller angles with distance; the horizon line contains all vanishing points of horizontal lines.

One-point perspective: front face parallel to picture plane; only depth lines converge to the single principal vanishing point.

Two-point perspective: verticals stay vertical; the two horizontal vanishing points lie on the horizon; their positions depend on the station point relative to the picture plane.

Cone of vision: the picture plane is placed so the object fits within a 60° cone (about 30° either side of the visual axis) to avoid distortion.

Measuring points: for a horizontal line with vanishing point V, its measuring point M lies on the horizon with station point S such that angle VSM equals the angle between the line and the picture plane.

ISO A-series sheet ratio is 1:√2; A0 = 1 m²; each smaller size halves the larger area (A1 = A0/2, etc.).

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

Dimension elements: dimension line with arrowheads at both ends, extension lines starting about 1 mm from the object, figures placed above the dimension line and readable from the bottom or right.

Scale notation: drawing scale = drawing length / real length; a 1:50 plan means 1 unit on paper equals 50 real units.

First-angle projection (ISO): the object is between observer and plane; the view from the front is placed above the plan, and the left-side view appears on the right.

Third-angle projection (common in US practice): the plane is between observer and object; the top view is placed above the front view.

Tolerances and fits: ISO tolerance grades IT01–IT18; a hole basis system uses H for the hole and letters a–zc for the shaft.

Section views: cutting plane indicated by a chain line with arrows; hatched areas at 45° thin lines represent cut material; adjacent parts are hatched in opposite directions.

Rule of thirds: key elements placed on the lines dividing the frame into thirds produce balanced tension; intersections are strong focal points.

Golden rectangle: sides in ratio 1:1.618; removing a square leaves a smaller golden rectangle, generating a logarithmic spiral.

Root rectangles: √2 (A-series), √3, √4 (double square) and √5 rectangles are constructed by swinging diagonals from a square.

Dynamic symmetry: a rectangle's diagonal and its reciprocal diagonal define a grid of proportional subdivisions usable for layout.

Visual weight: larger, darker, more saturated, or higher-contrast elements read as heavier; balance is achieved by distributing weight around the optical centre (slightly above geometric centre).

Gestalt grouping: proximity, similarity, continuity and closure determine how separate marks are read as one figure.

Modular proportion: a module repeated by a ratio (e.g., 2:3, 3:5, 5:8 Fibonacci) produces consistent rhythm across a composition.

Contrast hierarchy: value contrast dominates at a distance; hue and saturation contrast dominate at close range.

Project documentation set: sketch, dimensioned orthographic views, axonometric or perspective view, and a scale model or mock-up.

Scale selection rule: 1:1 for small objects, 1:5 to 1:20 for furniture, 1:50 to 1:200 for interiors and buildings.

Standard human dimensions used in design: eye level 1.60–1.70 m, comfortable seat height 0.42–0.45 m, table height 0.72–0.75 m.

Modular coordination: base module 100 mm (M = 100), with multi-modules (3M, 6M, 12M) used for building components and furniture grids.

Material representation: section hatching distinguishes materials; wood grain lines, metal at 45° fine lines, concrete stipple, glass with thin parallel lines.

Prototyping sequence: orthographic drawings → axonometric for spatial check → scale model → full-size mock-up for ergonomic verification.

Cost and cutting optimisation: nesting parts on sheet stock by aligning grain direction and minimising offcuts; sheet area divided by part area gives the yield ratio.

Presentation drawing: perspective view with a chosen station point and horizon matched to the viewer's eye level gives the most realistic impression for client approval.

Thales' theorem: a line parallel to one side of a triangle cutting the other two sides divides them in proportional segments; AB/BC = A'B'/B'C'.

Mean proportional (geometric mean) of segments a and b: height h of a right triangle satisfies h² = a·b; construct by drawing a semicircle on a+b and erecting a perpendicular at the junction point.

Golden section of segment AB: divide so that AB/AP = AP/PB ≈ 1.618; construction places AP as the side of a square plus the diagonal remainder.

Regular polygon inscribed in a circle: central angle = 360°/n; side length s = 2R·sin(180°/n); apothem a = R·cos(180°/n).

Approximate construction of any regular n-gon: divide a diameter into n equal parts, draw arcs from the diameter endpoints through the second division point; intersections give the vertices.

Tangent to a circle from external point P: the circle with diameter OP (O = centre) intersects the given circle at the two tangency points.

Ellipse as locus: sum of distances to foci is constant (2a); major axis 2a, minor axis 2b, focal distance c with a² = b² + c².

Parabola: points equidistant from focus F and directrix d; vertex is the midpoint between F and d; axis is perpendicular to d through F.

Hyperbola: absolute difference of distances to foci is constant 2a; asymptotes pass through the centre with slope b/a where c² = a² + b².

Dihedral system: horizontal plane H and vertical plane V meet at the ground line; the horizontal projection lies on H, the vertical projection on V, related by a vertical reference line perpendicular to the ground line.

Point coordinates: (x, y, z) where x = distance along the ground line, y = depth (distance to V), z = height (distance to H); in the folded view y appears below the ground line and z above it.

A line is true length in a view when it is parallel to that projection plane; if parallel to H it shows true length in plan and true angle in elevation only when also parallel to V.

A plane appears in true shape in the view perpendicular to its normal; edge-on (degenerate) in the view parallel to its normal.

Change of plane method: draw a new reference line perpendicular to the projection of a line to obtain its true length in one step.

Intersection of two planes: find two points common to both, typically where each plane's traces meet; the line through them is the intersection.

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

Visibility rule: in a view, the element whose depth (or height) coordinate is closer to the observer occludes the other at a crossing.

Isometric projection: three axes at 120°; foreshortening equal on all three; true isometric drawing uses full scale along axes (no 0.816 reduction).

Dimetric and trimetric axonometry differ by the chosen angles between axes; scale factors on each axis follow the chosen projection direction.

Cavalier and cabinet oblique: front face drawn true shape; receding axis at 45° (cabinet uses half scale on the receding axis).

Perspective: objects of equal size subtend smaller angles with distance; the horizon line contains all vanishing points of horizontal lines.

One-point perspective: front face parallel to picture plane; only depth lines converge to the single principal vanishing point.

Two-point perspective: verticals stay vertical; the two horizontal vanishing points lie on the horizon; their positions depend on the station point relative to the picture plane.

Cone of vision: the picture plane is placed so the object fits within a 60° cone (about 30° either side of the visual axis) to avoid distortion.

Measuring points: for a horizontal line with vanishing point V, its measuring point M lies on the horizon with station point S such that angle VSM equals the angle between the line and the picture plane.

ISO A-series sheet ratio is 1:√2; A0 = 1 m²; each smaller size halves the larger area (A1 = A0/2, etc.).

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

Dimension elements: dimension line with arrowheads at both ends, extension lines starting about 1 mm from the object, figures placed above the dimension line and readable from the bottom or right.

Scale notation: drawing scale = drawing length / real length; a 1:50 plan means 1 unit on paper equals 50 real units.

First-angle projection (ISO): the object is between observer and plane; the view from the front is placed above the plan, and the left-side view appears on the right.

Third-angle projection (common in US practice): the plane is between observer and object; the top view is placed above the front view.

Tolerances and fits: ISO tolerance grades IT01–IT18; a hole basis system uses H for the hole and letters a–zc for the shaft.

Section views: cutting plane indicated by a chain line with arrows; hatched areas at 45° thin lines represent cut material; adjacent parts are hatched in opposite directions.

Rule of thirds: key elements placed on the lines dividing the frame into thirds produce balanced tension; intersections are strong focal points.

Golden rectangle: sides in ratio 1:1.618; removing a square leaves a smaller golden rectangle, generating a logarithmic spiral.

Root rectangles: √2 (A-series), √3, √4 (double square) and √5 rectangles are constructed by swinging diagonals from a square.

Dynamic symmetry: a rectangle's diagonal and its reciprocal diagonal define a grid of proportional subdivisions usable for layout.

Visual weight: larger, darker, more saturated, or higher-contrast elements read as heavier; balance is achieved by distributing weight around the optical centre (slightly above geometric centre).

Gestalt grouping: proximity, similarity, continuity and closure determine how separate marks are read as one figure.

Modular proportion: a module repeated by a ratio (e.g., 2:3, 3:5, 5:8 Fibonacci) produces consistent rhythm across a composition.

Contrast hierarchy: value contrast dominates at a distance; hue and saturation contrast dominate at close range.

Project documentation set: sketch, dimensioned orthographic views, axonometric or perspective view, and a scale model or mock-up.

Scale selection rule: 1:1 for small objects, 1:5 to 1:20 for furniture, 1:50 to 1:200 for interiors and buildings.

Standard human dimensions used in design: eye level 1.60–1.70 m, comfortable seat height 0.42–0.45 m, table height 0.72–0.75 m.

Modular coordination: base module 100 mm (M = 100), with multi-modules (3M, 6M, 12M) used for building components and furniture grids.

Material representation: section hatching distinguishes materials; wood grain lines, metal at 45° fine lines, concrete stipple, glass with thin parallel lines.

Prototyping sequence: orthographic drawings → axonometric for spatial check → scale model → full-size mock-up for ergonomic verification.

Cost and cutting optimisation: nesting parts on sheet stock by aligning grain direction and minimising offcuts; sheet area divided by part area gives the yield ratio.

Presentation drawing: perspective view with a chosen station point and horizon matched to the viewer's eye level gives the most realistic impression for client approval.