In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity , a number ranging from to .
A drawing board is, in its antique form, a kind of multipurpose desk which can be used for any kind of drawing, writing or impromptu sketching on a large sheet of paper or for reading a large format book or other oversized document or for drafting precise technical illustrations. The drawing table used to be a frequent companion to a pedestal desk in a study or private library, during the pre-industrial and early industrial era.
In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an idealized ruler and a pair of compasses.
The Archimedean spiral (also known as Archimedes' spiral, the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. The term Archimedean spiral is sometimes used to refer to the more general class of spirals of this type (see below), in contrast to Archimedes' spiral (the specific arithmetic spiral of Archimedes). It is the locus corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line that rotates with constant angular velocity. Equivalently, in polar coordinates (r, θ) it can be described by the equation with real number b. Changing the parameter b controls the distance between loops.
A ruler, sometimes called a rule, scale or a line gauge, is an instrument used to make length measurements, whereby a user estimates a length by reading from a series of markings called "rules" along an edge of the device. Commonly the instrument is rigid and the edge itself is a straightedge, which additionally allows one to draw straight lines. Some rulers, such as cloth or paper tape measures, are non-rigid. Specialty rulers exist that have flexible edges that retain a chosen shape; these find use in sewing, arts, and crafts.
In geometry, two geometric objects are perpendicular if their intersection forms right angles at the point of intersection called a foot. The condition of perpendicularity may be represented graphically using the perpendicular symbol, ⟂. Perpendicular intersections can happen between two lines, between a line and a plane, and between two planes.
Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and a compass.
In the branch of mathematics known as Euclidean geometry, the Poncelet–Steiner theorem is one of several results concerning compass and straightedge constructions having additional restrictions imposed on the traditional rules. This result, related to the rusty compass equivalence, states that whatever can be constructed by straightedge and compass together can be constructed by straightedge alone, provided that a single circle and its centre are given:
Caliper(s) or calliper(s) are an instrument used to measure the dimensions of an object or hole; namely, the length, width, thickness, diameter or depth of an object or hole. The word caliper comes from latin roots meaning precise pincer.
A milliradian is an SI derived unit for angular measurement which is defined as a thousandth of a radian (0.001 radian). Milliradians are used in adjustment of firearm sights by adjusting the angle of the sight compared to the barrel. Milliradians are also used for comparing shot groupings, or to compare the difficulty of hitting different sized shooting targets at different distances. When using a scope with both mrad adjustment and a reticle with mrad markings, the shooter can use the reticle as a ruler to count the number of mrads a shot was off-target, which directly translates to the sight adjustment needed to hit the target with a follow-up shot. Optics with mrad markings in the reticle can also be used to make a range estimation of a known size target, or vice versa, to determine a target size if the distance is known, a practice called "milling".
Marking out or layout means the process of transferring a design or pattern to a workpiece, as the first step in the manufacturing process. It is performed in many industries or hobbies although in the repetition industries the machine's initial setup is designed to remove the need to mark out every individual piece.
The sector, also known as a sector rule, proportional compass, or military compass, was a major calculating instrument in use from the end of the sixteenth century until the nineteenth century. It is an instrument consisting of two rulers of equal length joined by a hinge. A number of scales are inscribed upon the instrument which facilitate various mathematical calculations. It was used for solving problems in proportion, multiplication and division, geometry, and trigonometry, and for computing various mathematical functions, such as square roots and cube roots. Its several scales permitted easy and direct solutions of problems in gunnery, surveying and navigation. The sector derives its name from the fourth proposition of the sixth book of Euclid, where it is demonstrated that similar triangles have their like sides proportional. Some sectors also incorporated a quadrant, and sometimes a clamp at the end of one leg which allowed the device to be used as a gunner's quadrant.
A beam compass is a compass with a beam and sliding sockets or cursors for drawing and dividing circles larger than those made by a regular pair of compasses. The instrument can be as a whole, or made on the spot with individual sockets and any suitable beam.
Drafting tools may be used for measurement and layout of drawings, or to improve the consistency and speed of creation of standard drawing elements. Tools such as pens and pencils mark the drawing medium. Other tools such as straight edges, assist the operator in drawing straight lines, or assist the operator in drawing complicated shapes repeatedly. Various scales and the protractor are used to measure the lengths of lines and angles, allowing accurate scale drawing to be carried out. The compass is used to draw arcs and circles. A drawing board was used to hold the drawing media in place; later boards included drafting machines that sped the layout of straight lines and angles. Tools such as templates and lettering guides assisted in the drawing of repetitive elements such as circles, ellipses, schematic symbols and text. Other auxiliary tools were used for special drawing purposes or for functions related to the preparation and revision of drawings. The tools used for manual technical drawing have been displaced by the advent of computer-aided drawing, drafting and design (CADD).
An ellipsograph is a mechanism that generates the shape of an ellipse. One common form of ellipsograph is known as the trammel of Archimedes. It consists of two shuttles which are confined to perpendicular channels or rails and a rod which is attached to the shuttles by pivots at adjustable positions along the rod.
A cyclograph is an instrument for drawing arcs of large diameter circles whose centres are inconveniently or inaccessibly located, one version of which was invented by Scottish architect and mathematician Peter Nicholson.
The quadratrix or trisectrix of Hippias is a curve which is created by a uniform motion. It is one of the oldest examples for a kinematic curve. Its discovery is attributed to the Greek sophist Hippias of Elis, who used it around 420 BC in an attempt to solve the angle trisection problem. Later around 350 BC Dinostratus used it in an attempt to solve the problem of squaring the circle.
A schema for horizontal dials is a set of instructions used to construct horizontal sundials using compass and straightedge construction techniques, which were widely used in Europe from the late fifteenth century to the late nineteenth century. The common horizontal sundial is a geometric projection of an equatorial sundial onto a horizontal plane.
Geometric Constructions is a mathematics textbook on constructible numbers, and more generally on using abstract algebra to model the sets of points that can be created through certain types of geometric construction, and using Galois theory to prove limits on the constructions that can be performed. It was written by George E. Martin, and published by Springer-Verlag in 1998 as volume 81 of their Undergraduate Texts in Mathematics book series.
Keuffel & Esser Arrow beam compass set for drafting.
18th-century ellipse-drawing compass (MHS Geneva)
Simple reduction compass (MHS Geneva).
Sliding-pivot reduction compass by Nairne on London, 18th century (MHS Geneva).
