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56-401: The term originally referred to portions of a log that had been riven (split lengthwise) into four or six sections. The radial members of a wagon wheel were made by carving a spoke (from a log) into their finished shape. A spokeshave is a tool originally developed for this purpose. Eventually, the term spoke was more commonly applied to the finished product of the wheelwright 's work than to

112-497: A of 67.5°. Regarding r 3 : The size of the spoke holes in the flange does not matter for the needed spoke length. This term removes the effect of the hole size. Since the holes are usually small (just over 2 mm in diameter), the effect is small and in practice matters little. For radially spoked wheels (zero crossings), the formula simplifies to the Pythagorean theorem , with spoke length l plus r 3 being

168-437: A "Z" to keep it from pulling through its hole in the hub. The bent version has the advantage of replacing a broken spoke in a rear bicycle wheel without having to remove the rear gears . Wire wheels, with their excellent weight-to-strength ratio, soon became popular for light vehicles. For everyday cars, wire wheels were soon replaced by the less expensive metal disc wheel, but wire wheels remained popular for sports cars up to

224-561: A circle in book III of the Elements (c. 300 BC). In Apollonius ' work Conics (c. 225 BC) he defines a tangent as being a line such that no other straight line could fall between it and the curve . Archimedes (c.  287 – c.  212 BC) found the tangent to an Archimedean spiral by considering the path of a point moving along the curve. In the 1630s Fermat developed the technique of adequality to calculate tangents and other problems in analysis and used this to calculate tangents to

280-513: A circle is always normal to the circle itself. These methods led to the development of differential calculus in the 17th century. Many people contributed. Roberval discovered a general method of drawing tangents, by considering a curve as described by a moving point whose motion is the resultant of several simpler motions. René-François de Sluse and Johannes Hudde found algebraic algorithms for finding tangents. Further developments included those of John Wallis and Isaac Barrow , leading to

336-510: A good quality spoke is capable of supporting about 225 kgf (c. 500 pounds-force or 2,200 newtons ) of tension, they are used at a fraction of this load to avoid suffering fatigue failures. Since bicycle and wheelchair wheel spokes are only in tension, flexible and strong materials such as synthetic fibers, are also occasionally used. Metal spokes can also be ovalized or bladed to reduce aerodynamic drag, and butted (double or even triple) to reduce weight while maintaining strength. A variation on

392-404: A point if they have the same tangent at a point, and orthogonal if their tangent lines are orthogonal. The formulas above fail when the point is a singular point . In this case there may be two or more branches of the curve that pass through the point, each branch having its own tangent line. When the point is the origin, the equations of these lines can be found for algebraic curves by factoring

448-435: A tangent line at the origin that is vertical. The graph y = x illustrates another possibility: this graph has a cusp at the origin. This means that, when h approaches 0, the difference quotient at a = 0 approaches plus or minus infinity depending on the sign of x . Thus both branches of the curve are near to the half vertical line for which y =0, but none is near to the negative part of this line. Basically, there

504-457: A width of r 1 sin( a ). Equivalently, the law of cosines may be used to first compute the length of the spoke as projected on the wheel's plane (as illustrated in the diagram), followed by an application of the Pythagorean theorem . Riving Wood splitting ( riving , cleaving) is an ancient technique used in carpentry to make lumber for making wooden objects, some basket weaving , and to make firewood . Unlike wood sawing ,

560-468: Is and it follows that the equation of the normal line at (X, Y) is Similarly, if the equation of the curve has the form f ( x ,  y ) = 0 then the equation of the normal line is given by If the curve is given parametrically by then the equation of the normal line is The angle between two curves at a point where they intersect is defined as the angle between their tangent lines at that point. More specifically, two curves are said to be tangent at

616-416: Is called the point of tangency . The tangent line is said to be "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point. The tangent line to a point on a differentiable curve can also be thought of as a tangent line approximation , the graph of the affine function that best approximates the original function at the given point. Similarly,

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672-425: Is denoted by g ( x ) {\displaystyle g(x)} , then the equation of the tangent line is given by When the equation of the curve is given in the form f ( x ,  y ) = 0 then the value of the slope can be found by implicit differentiation , giving The equation of the tangent line at a point ( X , Y ) such that f ( X , Y ) = 0 is then This equation remains true if in which case

728-407: Is no tangent at the origin in this case, but in some context one may consider this line as a tangent, and even, in algebraic geometry , as a double tangent . The graph y = | x | of the absolute value function consists of two straight lines with different slopes joined at the origin. As a point q approaches the origin from the right, the secant line always has slope 1. As a point q approaches

784-401: Is that the wood is much stronger. Due to this, it was historically used for building ships (e.g. drekars ) and traditional skis . A defining feature of shakes , which are like shingles, is that they are split rather than sawn and because the cell structure of the wood remains intact may be more durable, and similarly trunnels when split are stronger than when sawn. Sometimes wood splitting

840-515: Is typically so twisted it's near impossible to get a clean split, and elm. Any type of wood, being thick or tall, having large knots or twisted grain can make it difficult to split. In some cases, it is easiest to aim for the edges and split the log into multiple pieces. Batoning is splitting small pieces of wood for kindling or other purposes sometimes with a batoning chisel , a special chisel with one sharp side used for splitting. The advantages of splitting wood along its grain, rather than sawing it

896-502: Is undesirable. Methods to prevent splitting in woodworking are the butterfly joint , truss connector plates , or metal straps. Columns may be hollowed in the center to prevent splitting. Nail points may be blunted or pilot holes drilled to prevent splitting of lumber while nailing or screwing. End grain sealers are liquid products usually containing wax which helps prevent rapid drying of the ends of lumber resulting in splits. Metal end plates or S-shaped pieces of metal may be driven into

952-400: The artillery type were normally used. In a simple wooden wheel, a load on the hub causes the wheel rim to flatten slightly against the ground as the lowermost wooden spoke shortens and compresses. The other wooden spokes show no significant change. Wooden spokes are mounted radially . They are also dished, usually to the outside of the vehicle, to prevent wobbling. Also, the dishing allows

1008-475: The tangent plane to a surface at a given point is the plane that "just touches" the surface at that point. The concept of a tangent is one of the most fundamental notions in differential geometry and has been extensively generalized; see Tangent space . The word "tangent" comes from the Latin tangere , "to touch". Euclid makes several references to the tangent ( ἐφαπτομένη ephaptoménē ) to

1064-420: The 1960s. Spoked wheels are still popular on motorcycles and bicycles . When building a bicycle wheel , the spokes must have the correct length, otherwise there may not be enough threads engaged, producing a weaker wheel, or they may protrude through the rim and possibly puncture the inner tube. For bicycle spokes, the spoke length is defined from the flange seat to the thread tip. For spokes with bent ends,

1120-408: The butt ends of a timber. Splitting is the primary reason building codes do not allow notching in the bottom of joists and beams. Tangent In geometry , the tangent line (or simply tangent ) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on

1176-504: The curve, Euler's theorem implies ∂ g ∂ x ⋅ X + ∂ g ∂ y ⋅ Y + ∂ g ∂ z ⋅ Z = n g ( X , Y , Z ) = 0. {\displaystyle {\frac {\partial g}{\partial x}}\cdot X+{\frac {\partial g}{\partial y}}\cdot Y+{\frac {\partial g}{\partial z}}\cdot Z=ng(X,Y,Z)=0.} It follows that

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1232-420: The curve. More precisely, a straight line is tangent to the curve y = f ( x ) at a point x = c if the line passes through the point ( c , f ( c )) on the curve and has slope f ' ( c ) , where f ' is the derivative of f . A similar definition applies to space curves and curves in n -dimensional Euclidean space . The point where the tangent line and the curve meet or intersect

1288-418: The curve; in modern terminology, this is expressed as: the tangent to a curve at a point P on the curve is the limit of the line passing through two points of the curve when these two points tends to P . The intuitive notion that a tangent line "touches" a curve can be made more explicit by considering the sequence of straight lines ( secant lines ) passing through two points, A and B , those that lie on

1344-407: The derivative is evaluated at x = X {\displaystyle x=X} . When the curve is given by y = f ( x ), the tangent line's equation can also be found by using polynomial division to divide f ( x ) {\displaystyle f\,(x)} by ( x − X ) 2 {\displaystyle (x-X)^{2}} ; if the remainder

1400-405: The development of calculus in the 17th century. In the second book of his Geometry , René Descartes said of the problem of constructing the tangent to a curve, "And I dare say that this is not only the most useful and most general problem in geometry that I know, but even that I have ever desired to know". Suppose that a curve is given as the graph of a function , y = f ( x ). To find

1456-445: The direction in which "point B " approaches the vertex. At most points, the tangent touches the curve without crossing it (though it may, when continued, cross the curve at other places away from the point of tangent). A point where the tangent (at this point) crosses the curve is called an inflection point . Circles , parabolas , hyperbolas and ellipses do not have any inflection point, but more complicated curves do have, like

1512-449: The existing Mediterranean peoples to give rise, eventually, to classical Greece after the breaking of Minoan dominance and consolidations led by pre-classical Sparta and Athens . Neo- Chalcolithic /proto-historic period (1800-1200 BCE) paintings in various regions of India such as Chibbar Nulla, Chhatur Bhoj Nath Nulla, Kathotia, etc. depict the usage of chariots with spoked wheels. Celtic chariots introduced an iron rim around

1568-416: The function curve. The tangent at A is the limit when point B approximates or tends to A . The existence and uniqueness of the tangent line depends on a certain type of mathematical smoothness, known as "differentiability." For example, if two circular arcs meet at a sharp point (a vertex) then there is no uniquely defined tangent at the vertex because the limit of the progression of secant lines depends on

1624-408: The graph does not have a break or a sharp edge at p and it is neither plumb nor too wiggly near p . Then there is a unique value of k such that, as h approaches 0, the difference quotient gets closer and closer to k , and the distance between them becomes negligible compared with the size of h , if h is small enough. This leads to the definition of the slope of the tangent line to the graph as

1680-446: The graph of a cubic function , which has exactly one inflection point, or a sinusoid, which has two inflection points per each period of the sine . Conversely, it may happen that the curve lies entirely on one side of a straight line passing through a point on it, and yet this straight line is not a tangent line. This is the case, for example, for a line passing through the vertex of a triangle and not intersecting it otherwise—where

1736-465: The homogeneous equation of the tangent line is The equation of the tangent line in Cartesian coordinates can be found by setting z =1 in this equation. To apply this to algebraic curves, write f ( x ,  y ) as where each u r is the sum of all terms of degree r . The homogeneous equation of the curve is then Applying the equation above and setting z =1 produces as the equation of

Spoke - Misplaced Pages Continue

1792-415: The hub is rotated with reference to the rim one "angle between adjacent flange holes". Thus, multiplying the "angle between adjacent flange holes" by k gives the angle a . For example, a 32 spoke wheel has 16 spokes per side, 360° divided by 16 equals 22.5°. Multiply 22.5° ("angle between adjacent flange holes") by the number of crossings to get the angle a —if 3-crosses, the 32 spoke wheel has an angle

1848-469: The limit of the difference quotients for the function f . This limit is the derivative of the function f at x = a , denoted f  ′( a ). Using derivatives, the equation of the tangent line can be stated as follows: Calculus provides rules for computing the derivatives of functions that are given by formulas, such as the power function , trigonometric functions , exponential function , logarithm , and their various combinations. Thus, equations of

1904-447: The limits and derivatives to fail: either the geometric tangent exists, but it is a vertical line, which cannot be given in the point-slope form since it does not have a slope, or the graph exhibits one of three behaviors that precludes a geometric tangent. The graph y = x illustrates the first possibility: here the difference quotient at a = 0 is equal to h / h = h , which becomes very large as h approaches 0. This curve has

1960-605: The materials they used. The spoked wheel was invented to allow the construction of lighter and swifter vehicles. The earliest physical evidence for spoked wheels were found in the Sintashta culture , dating to c. 2000 BCE. Soon after this, horse cultures of the Caucasus region used horse-drawn spoked-wheel war chariots for the greater part of three centuries. They moved deep into the Greek peninsula , where they joined with

2016-409: The nominal spoke length does not include the width of the spoke at the bent end. For wheels with crossed spokes (which are the norm), the desired spoke length is where Regarding d : For a symmetric wheel such as a front wheel with no disc brake, this is half the distance between the flanges. For an asymmetric wheel such as a front wheel with disc brake or a rear wheel with chain derailleur ,

2072-456: The origin from the left, the secant line always has slope −1. Therefore, there is no unique tangent to the graph at the origin. Having two different (but finite) slopes is called a corner . Finally, since differentiability implies continuity, the contrapositive states discontinuity implies non-differentiability. Any such jump or point discontinuity will have no tangent line. This includes cases where one slope approaches positive infinity while

2128-413: The other approaches negative infinity, leading to an infinite jump discontinuity When the curve is given by y = f ( x ) then the slope of the tangent is d y / d x , {\displaystyle dy/dx,} so by the point–slope formula the equation of the tangent line at ( X ,  Y ) is where ( x ,  y ) are the coordinates of any point on the tangent line, and where

2184-402: The parabola. The technique of adequality is similar to taking the difference between f ( x + h ) {\displaystyle f(x+h)} and f ( x ) {\displaystyle f(x)} and dividing by a power of h {\displaystyle h} . Independently Descartes used his method of normals based on the observation that the radius of

2240-414: The rim to the brake in the opposite direction—(via the hub) when braking. Constructing a tension-spoked wheel from its constituent parts is called wheelbuilding and requires the correct building procedure for a strong and long-lasting end product. Tensioned spokes are usually attached to the rim or sometimes the hub with a spoke nipple . The other end is commonly peened into a disk or uncommonly bent into

2296-430: The slope of the tangent is infinite. If, however, the tangent line is not defined and the point ( X , Y ) is said to be singular . For algebraic curves , computations may be simplified somewhat by converting to homogeneous coordinates . Specifically, let the homogeneous equation of the curve be g ( x ,  y ,  z ) = 0 where g is a homogeneous function of degree n . Then, if ( X ,  Y ,  Z ) lies on

Spoke - Misplaced Pages Continue

2352-440: The slope, r 2 minus r 1 being the base, and d being the rise: The spoke length formula computes the length of the space diagonal of an imaginary rectangular box . Imagine holding a wheel in front of you such that a nipple is at the top. Look at the wheel from along the axis. The spoke through the top hole is now a diagonal of the imaginary box. The box has a depth of d , a height of r 2 - r 1 cos( α ) and

2408-409: The tangent line at the point p = ( a , f ( a )), consider another nearby point q = ( a + h , f ( a + h )) on the curve. The slope of the secant line passing through p and q is equal to the difference quotient As the point q approaches p , which corresponds to making h smaller and smaller, the difference quotient should approach a certain limiting value k , which is the slope of

2464-399: The tangent line at the point p . If k is known, the equation of the tangent line can be found in the point-slope form: To make the preceding reasoning rigorous, one has to explain what is meant by the difference quotient approaching a certain limiting value k . The precise mathematical formulation was given by Cauchy in the 19th century and is based on the notion of limit . Suppose that

2520-424: The tangent line does not exist for the reasons explained above. In convex geometry , such lines are called supporting lines . The geometrical idea of the tangent line as the limit of secant lines serves as the motivation for analytical methods that are used to find tangent lines explicitly. The question of finding the tangent line to a graph, or the tangent line problem, was one of the central questions leading to

2576-411: The tangent line is not defined. However, it may occur that the tangent line exists and may be computed from an implicit equation of the curve. The line perpendicular to the tangent line to a curve at the point of tangency is called the normal line to the curve at that point. The slopes of perpendicular lines have product −1, so if the equation of the curve is y = f ( x ) then slope of the normal line

2632-432: The tangent line. The equation in this form is often simpler to use in practice since no further simplification is needed after it is applied. If the curve is given parametrically by then the slope of the tangent is giving the equation for the tangent line at t = T , X = x ( T ) , Y = y ( T ) {\displaystyle \,t=T,\,X=x(T),\,Y=y(T)} as If

2688-408: The tangents to graphs of all these functions, as well as many others, can be found by the methods of calculus. Calculus also demonstrates that there are functions and points on their graphs for which the limit determining the slope of the tangent line does not exist. For these points the function f is non-differentiable . There are two possible reasons for the method of finding the tangents based on

2744-421: The theory of Isaac Newton and Gottfried Leibniz . An 1828 definition of a tangent was "a right line which touches a curve, but which when produced, does not cut it". This old definition prevents inflection points from having any tangent. It has been dismissed and the modern definitions are equivalent to those of Leibniz , who defined the tangent line as the line through a pair of infinitely close points on

2800-441: The value of d is different for the left and right sides. a is the angle between (1) the radius to a nipple hole in the rim to which a spoke is attached, and, (2) the radius to the flange hole holding the spoke. The spoke crosses either 1, 2, or 3 oppositely pointing spokes depending on the lacing design. On the flange, the angle between the radii of adjacent holes is 360°/ m (for equally spaced holes). For each spoke crossed,

2856-445: The wheel in the 1st millennium BCE. The spoked wheel was in continued use without major modification until the 1870s, when wire wheels and rubber tires were invented. Spokes can be made of wood, metal, or synthetic fiber depending on whether they will be in tension or compression . The original type of spoked wheel with wooden spokes was used for horse -drawn carriages and wagons . In early motor cars, wooden spoked wheels of

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2912-673: The wheel to compensate for expansion of the spokes due to absorbed moisture by dishing more. For use in bicycles , heavy wooden-spoked wheels were replaced by lighter wheels with spokes made of tensioned, adjustable metal wires, called wire wheels . These are also used in wheelchairs , motorcycles , automobiles , and early aircraft . Some types of wheels have removable spokes that can be replaced individually if they break or bend. These include bicycle and wheelchair wheels. High quality bicycles with conventional wheels use spokes of stainless steel , while cheaper bicycles may use galvanized (also called "rustless") or chrome plated spokes. While

2968-418: The wire-spoked wheel was Tioga's "Tension Disk", which appeared superficially to be a solid disk but was in fact constructed using the same principles as a normal tension-spoked wheel. Instead of individual wire spokes, a continuous thread of Kevlar ( aramid ) was used to lace the hub to the rim under high tension. The threads were encased in a translucent disk for protection and some aerodynamic benefit, but this

3024-550: The wood is split along the grain using tools such as a hammer and wedges , splitting maul , cleaving axe , side knife , or froe . In woodworking carpenters use a wooden siding which gets its name, clapboard, from originally being split from logs—the sound of the plank against the log being a clap. This is used in clapboard architecture and for wainscoting . Coopers use oak clapboards to make barrel staves. Split-rail fences are made with split wood. Some Native Americans traditionally make baskets from black ash by pounding

3080-557: The wood with a mallet and pulling long strips from the log. Log splitting is the act of splitting firewood from logs that have been pre-cut into sections (rounds, bolts, billets ). This can be done by hand, using an axe or maul , or by using a mechanical log splitter . When splitting a log by hand, it is best to aim for the cracks (called checks), if there are any visible. Some types of wood are harder to split than others, including extremely hard woods, as well as types like gum which an axe will often bounce off of, and cherry, which

3136-404: Was not a structural component. Wire spokes can be radial to the hub but are more often mounted tangentially to the hub. Tangential spoking allows for the transfer of torque between the rim and the hub. Tangential spokes are thus necessary for the drive wheel, which has torque at the hub from pedalling, and any wheels using hub-mounted brakes such as disk or band brakes, which transfer torque from

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