Straightedge

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A triangular architect's scale Architectural scale.jpg
A triangular architect's scale

A straightedge or straight edge is a tool used for drawing straight lines, or checking their straightness. If it has equally spaced markings along its length, it is usually called a ruler.

Ruler An instrument used to measure distances or to rule straight lines

A ruler, sometimes called a rule or line gauge, is a device used in geometry and technical drawing, as well as the engineering and construction industries, to measure or draw straight lines.

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Straightedges are used in the automotive service and machining industry to check the flatness of machined mating surfaces.

True straightness can in some cases be checked by using a laser line level as an optical straightedge: it can illuminate an accurately straight line on a flat surface such as the edge of a plank or shelf.

Laser line level

A laser line level is a tool combining a spirit level and/or plumb bob with a laser to display an accurately horizontal or vertical illuminated line on a surface the laser line level is laid against. Laser line levels are used wherever accurate verticals and horizontals are required, typically in the construction and cabinetry industries. Some models are inexpensive enough for do-it-yourself applications.

A pair of straightedges called winding sticks are used in woodworking to amplify twist (wind) in pieces of wood.

Winding stick

In woodworking and carpentry, a pair of winding sticks is a tool that aids in viewing twist or wind in pieces of lumber (timber) by amplifying the defect. Winding sticks can be as simple as any two straight sticks or they can be elegant, decorated, dimensionally stable wood like mahogany. A pair of framing squares may also be suitable. Traditionally they are 16 inches (41 cm) to 30 inches (76 cm) long 1.75 inches (4.4 cm) tall and tapered in their height from 38 inch (0.95 cm) to 18 inch (0.32 cm). The longer the winding sticks, the more they will amplify the wind. It is common for a woodworker to make a matching pair for the purpose, and contrasting colors of woods improve the discernability of differences in height and level between the two sticks as they are compared.

Compass-and-straightedge construction

An idealized straightedge is used in compass-and-straightedge constructions in plane geometry. It may be used:

Euclidean geometry mathematical system attributed to Euclid

Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The Elements begins with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, explained in geometrical language.

It may not be marked or used together with the compass so as to transfer the length of one segment to another.

It is possible to do all compass and straightedge constructions without the straightedge. That is, it is possible, using only a compass, to find the intersection of two lines given two points on each, and to find the tangent points to circles. It is not, however, possible to do all constructions using only a straightedge. It is possible to do them with straightedge alone given one circle and its center.

Poncelet–Steiner theorem theorem

In Euclidean geometry, the Poncelet–Steiner theorem is one of several results concerning compass and straightedge constructions with additional restrictions. This result 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.

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Angle figure formed by two rays; coordinate system in one-dimensional space or The amount of turn between two straight lines that have a common end point (the vertex).

In plane geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane. Angles are also formed by the intersection of two planes in Euclidean and other spaces. These are called dihedral angles. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the spherical angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles.

Circle simple curve of Euclidean geometry

A circle is a simple closed shape. It is the set of all points in a plane that are at a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves so that its distance from a given point is constant. The distance between any of the points and the centre is called the radius. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted.

Constructible number real number

In geometry and algebra, a real number r is constructible if and only if, given a line segment of unit length, a line segment of length |r| can be constructed with compass and straightedge in a finite number of steps. Not all real numbers are constructible and to describe those that are, algebraic techniques are usually employed. However, in order to employ those techniques, it is useful to first associate points with constructible numbers.

Straightedge and compass construction way of drawing geometric objects

Straightedge and compass construction, also known as ruler-and-compass construction or classical construction, is the construction of lengths, angles, and other geometric figures using only an idealized ruler and compass.

Perpendicular property of being perpendicular (perpendicularity) is the relationship between two lines which meet at a right angle (90 degrees). The property extends to other related geometric objects

In elementary geometry, the property of being perpendicular (perpendicularity) is the relationship between two lines which meet at a right angle. The property extends to other related geometric objects.

Doubling the cube geometric problem of constructing a cube with twice the volume of a given cube, not possible with compass and straightedge

Doubling the cube, also known as the Delian problem, is an ancient geometric problem. Given the edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first. As with the related problems of squaring the circle and trisecting the angle, doubling the cube is now known to be impossible using only a compass and straightedge, but even in ancient times solutions were known that employed other tools.

Angle trisection the construction of an angle equal to one third a given angle

Angle trisection is a classical problem of compass and straightedge constructions 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.

The art of origami or paper folding has received a considerable amount of mathematical study. Fields of interest include a given paper model's flat-foldability and the use of paper folds to solve mathematical equations.

The Huzita–Hatori axioms or Huzita–Justin axioms are a set of rules related to the mathematical principles of paper folding, describing the operations that can be made when folding a piece of paper. The axioms assume that the operations are completed on a plane, and that all folds are linear. These are not a minimal set of axioms but rather the complete set of possible single folds.

Poincaré half-plane model

In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H , together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry.

In mathematics, the Mohr–Mascheroni theorem states that any geometric construction that can be performed by a compass and straightedge can be performed by a compass alone.

Hand scraper single-edged tool used to scrape metal from a surface

A hand scraper is a single-edged tool used to scrape metal from a surface. This may be required where a surface needs to be trued, corrected for fit to a mating part, needs to retain oil, or to give a decorative finish.

Horocycle

In hyperbolic geometry, a horocycle is a curve whose normal or perpendicular geodesics all converge asymptotically in the same direction. It is the two-dimensional example of a horosphere.

The compass equivalence theorem is an important statement in compass and straightedge constructions. The tool advocated by Plato in these constructions is a divider or collapsing compass, that is, a compass that "collapses" whenever it is lifted from a page, so that it may not be directly used to transfer distances. The modern compass with its fixable aperture can be used to transfer distances directly and so appears to be a more powerful instrument. However, the compass equivalence theorem states that any construction via a "modern compass" may be attained with a collapsing compass. This can be shown by establishing that with a collapsing compass, given a circle in the plane, it is possible to construct another circle of equal radius, centered at any given point on the plane. This theorem is Proposition II of Book I of Euclid's Elements. The proof of this theorem has had a chequered history.

In Euclidean geometry, Apollonius' problem is to construct all the circles that are tangent to three given circles. Special cases of Apollonius' problem are those in which at least one of the given circles is a point or line, i.e., is a circle of zero or infinite radius. The nine types of such limiting cases of Apollonius' problem are to construct the circles tangent to:

  1. three points
  2. three lines
  3. one line and two points
  4. two lines and a point
  5. one circle and two points
  6. one circle, one line, and a point
  7. two circles and a point
  8. one circle and two lines
  9. two circles and a line
Chalk line

A chalk line or chalk box is a tool for marking long, straight lines on relatively flat surfaces, much farther than is practical by hand or with a straightedge. They may be used to lay out straight lines between two points, or vertical lines by using the weight of the line reel as a plumb line.

Pentagon shape with five sides

In geometry, a pentagon is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.

Technical drawing tool Tools and instruments used for accurate and precise manual draughting

Technical drawing tools include and are not limited to: pens, rulers, compasses, protractors and drawing utilities. Drafting tools may be used for measurement and layout of drawings, or to improve the consistency and speed of creation of standard drawing elements. The tools used for manual technical drawing have been displaced by the advent of the personal computer and its common utilization as the main tool in computer-aided drawing, draughting and design (CADD).

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.

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