PAU Technical Drawing II (Andalusia) flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Circumradius of equilateral triangle

For side a, R = a/√3 ≈ 0.577a. The circumcentre coincides with the incentre, centroid and orthocentre, all at one third of the height from the base.

Regular polygon central angle

The central angle subtended by one side of a regular n-gon is 360°/n. The interior angle is (n−2)·180°/n, and the exterior angle equals the central angle.

Golden ratio in pentagon

In a regular pentagon the diagonal divided by the side equals φ ≈ 1.618. This ratio lets you construct the pentagon from its side by mean-and-extreme section.

Right triangle altitude theorem

The altitude drawn to the hypotenuse is the geometric mean of the two segments it creates: h² = m·n. Each leg is the mean of the hypotenuse and its adjacent segment.

Rhombus diagonal relation

The diagonals of a rhombus are perpendicular bisectors of each other, so half-diagonals and side form a right triangle: (D/2)² + (d/2)² = a². Area is D·d/2.

Ellipse definition and elements

Locus of points whose distances to two foci add to a constant 2a. Semi-axes satisfy a² = b² + c², with c the focal half-distance; eccentricity e = c/a is below 1.

Hyperbola asymptotes

For a hyperbola with semi-axes a and b, the asymptotes pass through the centre with slopes ±b/a and are the diagonals of the central rectangle of sides 2a and 2b.

Parabola parameter

The parameter p is the distance from the focus to the directrix. The vertex lies midway, the axis is perpendicular to the directrix, and the latus rectum measures 2p.

Tangent to ellipse at point

The tangent at a point of an ellipse bisects the exterior angle formed by the two focal radii at that point; the normal bisects the interior angle.

Cone-plane conic rule

The conic depends on the cutting plane's inclination: perpendicular to the axis gives a circle, less inclined than the generatrix an ellipse, parallel to it a parabola, steeper a hyperbola.

Perpendicular bisector locus

The locus of points equidistant from two fixed points is the perpendicular bisector of the segment joining them; equidistance from two lines gives their angle bisectors.

Tangent from external point

Draw the circle with diameter OP, where O is the centre and P the external point. Its intersections with the given circle are the two tangency points.

Common external tangent construction

Centre an auxiliary circle at the larger centre with radius R−r, draw a tangent from the smaller centre, then extend that radius to the tangency point and draw the parallel radius.

Fillet centre in an angle

For an arc of radius r tangent to both sides of an angle θ, the centre lies on the bisector at distance r/sin(θ/2) from the vertex; tangency points are perpendicular feet.

Fillet between line and circle

Offset the line by r and draw a concentric circle of radius R+r (external link) or R−r (internal). Their intersections are the centres of the linking arcs.

Tangency point of two circles

When two circles are tangent, the point of contact lies on the line joining their centres; for internal tangency it lies on the extension of that line.

Thales proportionality theorem

A line parallel to one side of a triangle divides the other two sides proportionally: the segments on one side are to each other as the segments on the other.

Fourth proportional construction

To find x with a/b = c/x, draw two rays from a vertex, mark a and b on one and c on the other, join the ends of a and c, and draw a parallel through b.

Geometric mean construction

Lay segments m and n end to end, draw the semicircle on the total, and erect the perpendicular at the junction. Its length inside the semicircle is √(m·n).

Similar figure ratios

If two similar figures have linear ratio k, their areas are in ratio k² and their volumes in ratio k³; angles remain equal in both figures.

Scale definition

Scale is the ratio of drawing length to real length. At 1:50, one centimetre on paper represents 50 cm real, and areas are reduced by the factor 2500.

True length of a segment

Construct a right triangle whose legs are the horizontal projection and the difference of cotas. The hypotenuse is the true length of the segment in space.

Traces of a line

The horizontal trace is where the line meets the horizontal plane, found where its vertical projection crosses the ground line; the vertical trace is found analogously.

Line perpendicular to plane

A line is perpendicular to a plane when it is perpendicular to two lines of that plane, typically a horizontal and a frontal, giving right angles in true shape.

Line-plane intersection method

Pass an auxiliary projecting plane through the line, find its intersection line with the given plane, and the crossing of that line with the original line is the piercing point.

Euler polyhedron formula

For any convex polyhedron, vertices minus edges plus faces equals 2: V − E + F = 2. It holds for the five regular solids and all simple polyhedra.

Cone section types

A plane parallel to the base gives a circle, through the apex a triangle, oblique cutting all generatrices an ellipse, parallel to one generatrix a parabola, steeper a hyperbola.

Cone development angle

The lateral surface unfolds as a sector of radius equal to the generatrix g and angle α = 360°·r/g, where r is the base radius; the base circle is added tangent to the arc.

Isometric ellipse axes

A circle of diameter d in a coordinate plane projects as an ellipse with major axis 1.22d and minor axis 0.71d; the major axis is perpendicular to the normal axis.

Cavalier perspective rule

The frontal face is drawn true shape on the picture plane; the depth axis is drawn obliquely, usually at 45°, with a reduction coefficient commonly set to one half.

ISO A-series format

A0 measures 1 m² with sides in ratio 1:√2 (841×1189 mm). Each following size halves the larger side: A1 594×841, A2 420×594, A3 297×420, A4 210×297 mm.

Dimensioning conventions

Dimensions are given in millimetres without a unit symbol, written above the dimension line; extension lines start about 1–2 mm from the object and no dimension is repeated.