Andalusia PAU Technical Drawing Applied to Plastic Arts and Design II flashcards

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

Thales' proportionality theorem

Parallel lines cutting two transversals divide them into proportional segments, so AB/BC equals A'B'/B'C'. Its converse identifies parallelism from equal ratios.

Fourth proportional construction

Given a, b, c, draw two rays from a vertex, mark a and b on one and c on the other, then draw the parallel giving x with a/b = c/x, so x = bc/a.

Mean proportional

The value m satisfying a/m = m/b, hence m = √(ab). It equals the height to the hypotenuse in a right triangle with legs a and b.

Golden section ratio

A segment divided in golden section satisfies total/part = part/remainder = φ ≈ 1.618. Constructed with a perpendicular half-segment and an arc from its midpoint.

Homothety properties

With center O and ratio k, each point P maps to P' on ray OP with OP' = k·OP. Lines map to parallel lines; negative k inverts through O.

Similar figures area ratio

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

External tangency of circles

Two circles are externally tangent when the distance between centers equals the sum of radii; internally tangent when it equals the difference of radii.

Circle tangent to two lines

The center of any circle tangent to two intersecting lines lies on one of their angle bisectors; the radius equals the perpendicular distance to either line.

Tangent circle through a point

For a circle of radius r through P and tangent to a given circle, the center lies on a circle of radius r about P and on a concentric circle of radius R ± r.

Ellipse focal property

An ellipse is the locus of points whose distances to two foci sum to the constant 2a. The focal distance is c = √(a² − b²).

Parabola tangent bisector

The tangent at a point of a parabola bisects the angle between the focal radius and the line through the point parallel to the axis.

Hyperbola asymptotes

For a hyperbola with semiaxes a and b, the asymptotes pass through the center with slopes ±b/a, and c = √(a² + b²) locates the foci.

Cycloid arch dimensions

A cycloid generated by a circle of radius r rolling on a line has one arch of length 8r and area 3πr².

Line traces in dihedral

The horizontal and vertical traces of a line are where it pierces the two projection planes; both lie on the same projection perpendicular to the ground line.

Plane traces

A plane is represented by its intersections with the horizontal and vertical planes. The two traces meet on the ground line unless the plane is parallel to a projection plane.

Line piercing 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.

Intersection of two planes

Locate two points each belonging to both planes, typically by intersecting corresponding traces, then join them to obtain the intersection line.

True length by right triangle

The hypotenuse of a right triangle whose legs are the projected length of a segment and the difference of its depths equals the true length of the segment.

Theorem of three perpendiculars

If a line in a plane is perpendicular to the projection of an oblique line onto that plane, it is also perpendicular to the oblique line itself.

Distance point to plane

Drop the perpendicular from the point to the plane, find where it pierces the plane, then determine the true length of that perpendicular segment.

Euler's polyhedron formula

For any convex polyhedron, vertices minus edges plus faces equals two. A cube gives 8 − 12 + 6 = 2.

Sphere section radius

A plane at distance d from the center of a sphere of radius R cuts a circle of radius √(R² − d²); a plane through the center gives a great circle.

Cone lateral development

The lateral surface of a right cone unfolds into a circular sector of radius equal to the generatrix g and angle 360°·r/g, where r is the base radius.

Torus volume and surface

A torus generated by a circle of radius r whose center is R from the axis has volume 2π²Rr² and surface area 4π²Rr.

Isometric scale coefficient

In true isometric projection all three axes share the coefficient 0.816; isometric drawing rounds it to 1 for convenience.

Caballera reduction coefficient

In cavallera perspective the frontal plane is true size and depths are multiplied by a coefficient: 0.5 at 45°, about 0.816 at 30°, about 0.577 at 60°.

Isometric circle ellipse axes

A circle of diameter d lying in a coordinate plane projects isometrically as an ellipse with major axis 1.22d and minor axis 0.71d.

Axonometric axes angles

In isometric drawing the three axes meet at 120°. Dimetric and trimetric systems use unequal angles and unequal scale coefficients along the axes.

ISO A-series sheet ratio

A0 has an area of one square metre with sides in ratio √2; each following size halves the larger side, giving A4 as 210 by 297 millimetres.

Dimension line placement

Dimension lines run parallel to the measured edge with arrowheads at both ends, and their value is written above the line in true size whatever the scale.

First-angle projection symbol

First-angle projection places the view from the left on the right of the front view; the frame symbol is a truncated cone with its small circle away from the observer.

Section hatching convention

Cut surfaces are hatched with thin parallel lines usually at 45°. Shafts, bolts, and ribs cut lengthwise are left unhatched to avoid a false impression of solidity.