Straight Line and its construction
From Math Images
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  ImageIntro=Independently invented by a French army officer, CharlesNicolas Peaucellier and a Lithuanian (Nationality disputable; some say he is Russian.) mathematician Lipmann Lipkin, this is the first planar {{EasyBalloonLink=linkageBalloon=It is defined as a series of rigid links connected with joints to form a closed chain, or a series of closed chains. Each link has two or more joints, and the joints have various degrees of freedom to allow motion between the links.}} that drew a straight line without using a straight edge and it had important applications in engineering and mathematics.  +  ImageIntro=Independently invented by a French army officer, CharlesNicolas Peaucellier and a Lithuanian (Nationality disputable; some say he is Russian.) mathematician Lipmann Lipkin, this is the first planar {{EasyBalloonLink=linkageBalloon=It is defined as a series of rigid links connected with joints to form a closed chain, or a series of closed chains. Each link has two or more joints, and the joints have various degrees of freedom to allow motion between the links.}} that drew a straight line without using a straight edge and it had important applications in engineering and mathematics.'''<ref>Bryant, & Sangwin, 2008, p. 34</ref><ref>Kempe, 1877, p. 12</ref><ref>Taimina</ref>''' 
=Introduction=  =Introduction=  
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  For more properties on  +  For more properties on straight line, you can refer to the book '''''Experiencing Geometry''''' by David W. Henderson. 
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  {{!}}align="center"{{!}}[[Image:Straightline.jpgcenterborder400px]] '''Image 1'''{{!}}{{!}}align="center"{{!}}[[Image:SmallGreatCircles 700.gifcenterborder]]'''Image 2'''  +  {{!}}align="center"{{!}}[[Image:Straightline.jpgcenterborder400px]] '''Image 1'''{{!}}{{!}}align="center"{{!}}[[Image:SmallGreatCircles 700.gifcenterborder]]'''Image 2''' '''Weisstein''' 
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  {{!}}align="center"{{!}}'''Image 3'''  +  {{!}}align="center"{{!}}'''Image 3''' '''Bryant, & Sangwin, 2008, p. 18''' 
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{{!}}'''Image 3''' shows a patent drawing of an early steam engine. It is of the simplest form with a boiler (on the left), a cylinder with piston, a beam (on top) and a pump (on the right side) at the other end. The pump was usually used to extract water from the mines.  {{!}}'''Image 3''' shows a patent drawing of an early steam engine. It is of the simplest form with a boiler (on the left), a cylinder with piston, a beam (on top) and a pump (on the right side) at the other end. The pump was usually used to extract water from the mines.  
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  {{!}}{{HideShowThisShowMessage=Click here to show how this engine works.HideMessage=Click here to hide textHiddenText=When the piston is at its lowest position, steam is let into the cylinder from valve K and it pushes the piston upwards. Afterward, when the piston is at its highest position, cold water is let in from valve E, cooling the steam in the cylinder and causing the pressure in the the cylinder to drop below the atmospheric pressure. The difference in pressure caused the piston to move downwards. After the piston returns to the lowest position, the whole process is repeated. This kind of steam engine is called "atmospheric" because it utilized atmospheric pressure to cause the downward action of the piston (steam only balances out the atmospheric pressure and allow the piston to return to the highest point). Since in the downward motion, the piston pulls on the beam and in the upward motion, the beam pulls on the piston, the connection between the end of the piston rod and the beam is always in tension (under stretching) and that is why a chain is used as the connection.}}  +  {{!}}{{HideShowThisShowMessage=Click here to show how this engine works.HideMessage=Click here to hide textHiddenText=When the piston is at its lowest position, steam is let into the cylinder from valve K and it pushes the piston upwards. Afterward, when the piston is at its highest position, cold water is let in from valve E, cooling the steam in the cylinder and causing the pressure in the the cylinder to drop below the atmospheric pressure. The difference in pressure caused the piston to move downwards. After the piston returns to the lowest position, the whole process is repeated. This kind of steam engine is called "atmospheric" because it utilized atmospheric pressure to cause the downward action of the piston (steam only balances out the atmospheric pressure and allow the piston to return to the highest point). Since in the downward motion, the piston pulls on the beam and in the upward motion, the beam pulls on the piston, the connection between the end of the piston rod and the beam is always in tension (under stretching) and that is why a chain is used as the connection.'''Bryant, & Sangwin, 2008, p. 18''''''Wikipedia (Steam Engine)'''}} 
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  {{!}}Ideally, the piston moves in the vertical direction and the piston rod takes only axial loading, i.e. forces applied in the direction along the rod. However, from the above picture, it is clear that the end of the piston does not move in a straight line due to the fact that the end of the beam describes an  +  {{!}}Ideally, the piston moves in the vertical direction and the piston rod takes only axial loading, i.e. forces applied in the direction along the rod. However, from the above picture, it is clear that the end of the piston does not move in a straight line due to the fact that the end of the beam describes an arc of a circle. As a result, horizontal forces are created and subjected onto the piston rod. Consequently, the process of wear and tear is very much quickened and the efficiency of the engine greatly compromised. Now considering that the upanddown cycle repeats itself hundreds of times every minute and the engine is expected to run 24/7 to make profits for the investors, such defect in the engine must not be tolerated and thus poses a great need for improvements.'''Bryant, & Sangwin, 2008, p. 1821''' 
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{{!}}colspan="2"{{!}}Improvements were made. Firstly, "doubleaction" engines were made, part of which is shown in '''Image 4'''. Secondly, beam was dispensed and replaced by a gear as shown in '''Image 5'''. However, both of these methods were not satisfactory and the need for a linkage that produces straight line action was still imperative.  {{!}}colspan="2"{{!}}Improvements were made. Firstly, "doubleaction" engines were made, part of which is shown in '''Image 4'''. Secondly, beam was dispensed and replaced by a gear as shown in '''Image 5'''. However, both of these methods were not satisfactory and the need for a linkage that produces straight line action was still imperative.  
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  {{!}}align="center"{{!}}[[Image:Img325.gifcenterborder500px]]'''Image 4'''{{!}}{{!}}align="center"{{!}}[[Image:Img326.gifbordercenter200px]]'''Image 5'''  +  {{!}}align="center"{{!}}[[Image:Img325.gifcenterborder500px]]'''Image 4''' '''Bryant, & Sangwin, 2008, p. 1821'''{{!}}{{!}}align="center"{{!}}[[Image:Img326.gifbordercenter200px]]'''Image 5''' '''Bryant, & Sangwin, 2008, p. 1821''' 
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  {{!}}colspan="2"{{!}}{{HideShowThisShowMessage=Why those engines were unsatisfactory?HideMessage=HideHiddenText=In '''Image 4''', atmospheric pressure acts in both upward and downward strokes of the engine and two chains were used (one connected to the top of the arched end of the beam and one to the bottom), both of which will take turns to be in tension throughout one cycle. One might ask why chain was used all the time. The answer was simple: to fit the curved end of the beam. However, this does not fundamentally solved the problem and unfortunately created more. The additional chain increased the height of the engine and made the manufacturing very difficult (it was hard to make straight steel bars and rods back then) and costly. In '''Image 5''', after the beam was replaced by gear actions, the piston rod was fitted with teeth (labeled k) to drive the gear. Theoretically, this solves the problem fundamentally. The piston rod is confined between the guiding wheel at K and the gear, and it moves only in the upanddown motion. However, the practical problem was still there. The friction and the noise between all the guideways and the wheels could not be ignored, not to mention the increased possibility of failure and cost of maintenance due to additional parts.}}  +  {{!}}colspan="2"{{!}}{{HideShowThisShowMessage=Why those engines were unsatisfactory?HideMessage=HideHiddenText=In '''Image 4''', atmospheric pressure acts in both upward and downward strokes of the engine and two chains were used (one connected to the top of the arched end of the beam and one to the bottom), both of which will take turns to be in tension throughout one cycle. One might ask why chain was used all the time. The answer was simple: to fit the curved end of the beam. However, this does not fundamentally solved the problem and unfortunately created more. The additional chain increased the height of the engine and made the manufacturing very difficult (it was hard to make straight steel bars and rods back then) and costly. In '''Image 5''', after the beam was replaced by gear actions, the piston rod was fitted with teeth (labeled k) to drive the gear. Theoretically, this solves the problem fundamentally. The piston rod is confined between the guiding wheel at K and the gear, and it moves only in the upanddown motion. However, the practical problem was still there. The friction and the noise between all the guideways and the wheels could not be ignored, not to mention the increased possibility of failure and cost of maintenance due to additional parts.'''Bryant, & Sangwin, 2008, p. 1821'''}} 
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  {{!}}colspan="2"{{!}}James Watt found a mechanism that converted the linear motion of pistons in the cylinder to the semi circular motion of the beam (or the circular motion of the [http://en.wikipedia.org/wiki/Flywheel flywheel]) and vice versa. In 1784, he invented a [http://en.wikipedia.org/wiki/Linkage_(mechanical) three member linkage] that solved the linear motion to circular problem practically as illustrated by the animation below. In its simplest form, there are two radius arms that have the same lengths and a connecting arm with midpoint P. Point P moves in a straight line. However, this linkage only produced approximate straight line (a stretched figure 8 actually) as shown in '''Image 7''', much to the chagrin of the mathematicians who were after absolute straight lines. There is a more general form of the Watt's linkage that the two radius arms having different lengths like shown in '''Image 6'''. To make sure that Point P still move in the stretched figure 8, it has to be positioned such that it adheres to the ratio<math>\frac{AB}{CD} = \frac{CP}{CB}</math>.  +  {{!}}colspan="2"{{!}}James Watt found a mechanism that converted the linear motion of pistons in the cylinder to the semi circular motion of the beam (or the circular motion of the [http://en.wikipedia.org/wiki/Flywheel flywheel]) and vice versa. In 1784, he invented a [http://en.wikipedia.org/wiki/Linkage_(mechanical) three member linkage] that solved the linear motion to circular problem practically as illustrated by the animation below. In its simplest form, there are two radius arms that have the same lengths and a connecting arm with midpoint P. Point P moves in a straight line. However, this linkage only produced approximate straight line (a stretched figure 8 actually) as shown in '''Image 7''', much to the chagrin of the mathematicians who were after absolute straight lines. There is a more general form of the Watt's linkage that the two radius arms having different lengths like shown in '''Image 6'''. To make sure that Point P still move in the stretched figure 8, it has to be positioned such that it adheres to the ratio<math>\frac{AB}{CD} = \frac{CP}{CB}</math>.'''Bryant, & Sangwin, 2008, p. 24''' 
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{{!}}[[Image:Img327.gifcenterborder400px]]{{!}}{{!}}[[Image:Watts linkage.gifcenterborder]]  {{!}}[[Image:Img327.gifcenterborder400px]]{{!}}{{!}}[[Image:Watts linkage.gifcenterborder]]  
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  {{!}}align="center"{{!}}'''Image 6'''{{!}}{{!}}align="center"{{!}}'''Image 7'''  +  {{!}}align="center"{{!}}'''Image 6''' '''Bryant, & Sangwin, 2008, p. 23'''{{!}}{{!}}align="center"{{!}}'''Image 7''' '''Wikipedia (Watt's Linkage)''' 
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{{!}}[[Image:Watt2.gifcenterborder450px]]  {{!}}[[Image:Watt2.gifcenterborder450px]]  
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  {{!}}align="center"{{!}}'''Image 10'''  +  {{!}}align="center"{{!}}'''Image 10''' '''Lienhard, 1999, February 18''' 
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=='''The First Planar Straight Line Linkage  PeaucellierLipkin Linkage'''==  =='''The First Planar Straight Line Linkage  PeaucellierLipkin Linkage'''==  
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  {{!}}align="center"{{!}}[[Image:Peaucellier linkage animation.gifcenterborder]]'''Image 13'''{{!}}{{!}}Mathematicians and engineers had being searching for almost a century to find the solution to the straight line linkage but all had failed until 1864, a French army officer Charles Nicolas Peaucellier came up with his ''inversor linkage''. Interestingly, he did not publish his findings and proof until 1873, when Lipmann I. Lipkin, a student from University of St. Petersburg, demonstrated the same working model at the World Exhibition in Vienna. Peaucellier acknowledged Lipkin's independent findings with the publication of the details of his discovery in 1864 and the mathematical proof.  +  {{!}}align="center"{{!}}[[Image:Peaucellier linkage animation.gifcenterborder]]'''Image 13''' '''Wikipedia (Peaucellier–Lipkin linkage)'''{{!}}{{!}}Mathematicians and engineers had being searching for almost a century to find the solution to the straight line linkage but all had failed until 1864, a French army officer Charles Nicolas Peaucellier came up with his ''inversor linkage''. Interestingly, he did not publish his findings and proof until 1873, when Lipmann I. Lipkin, a student from University of St. Petersburg, demonstrated the same working model at the World Exhibition in Vienna. Peaucellier acknowledged Lipkin's independent findings with the publication of the details of his discovery in 1864 and the mathematical proof. '''Taimina''' 
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and <math> OC \cdot OP = ON \cdot OR</math>  and <math> OC \cdot OP = ON \cdot OR</math>  
  Therefore <math> ON = \frac {OC \cdot OP}{OR} = </math>constant, i.e. the length of <math>ON</math>(or the xcoordinate of <math>P</math> w.r.t <math>O</math>) does not change as points <math>C</math> and <math>P</math> move. Hence, point <math>P</math> moves in a straight line. ∎  +  Therefore <math> ON = \frac {OC \cdot OP}{OR} = </math>constant, i.e. the length of <math>ON</math>(or the xcoordinate of <math>P</math> w.r.t <math>O</math>) does not change as points <math>C</math> and <math>P</math> move. Hence, point <math>P</math> moves in a straight line. ∎'''Bryant, & Sangwin, 2008, p. 3336''' 
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{{!}}align="center"{{!}}'''Image 16'''  {{!}}align="center"{{!}}'''Image 16'''  
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  {{!}}The new linkage caused considerable excitement in London. Mr. Prim, "engineer to the House", utilized the new compact form invented by H.Hart to fit his new blowing engine which proved to be "exceptionally quiet in their operation." In this compact form, <math>DA=DC</math>, <math>AF=CF</math> and <math>AB = BC</math>. Point <math>E</math> and <math>F</math> are fixed pivots. In '''Image 16'''. F is the inversive center and points <math>D</math>,<math>F</math> and <math>B</math> are collinear and <math>DF \cdot DB</math> is of constant value. Mr. Prim's blowing engine used for ventilating the House of Commons, 1877. The crosshead of the reciprocating air pump is guided by a Peaucellier linkage shown in '''Image 17'''. The slatelined air cylinders had rubberflap inlet and exhaust valves and a piston whose periphery was formed by two rows of brush bristles. Prim's machine was driven by a steam engine.  +  {{!}}The new linkage caused considerable excitement in London. Mr. Prim, "engineer to the House", utilized the new compact form invented by H.Hart to fit his new blowing engine which proved to be "exceptionally quiet in their operation." In this compact form, <math>DA=DC</math>, <math>AF=CF</math> and <math>AB = BC</math>. Point <math>E</math> and <math>F</math> are fixed pivots. In '''Image 16'''. F is the inversive center and points <math>D</math>,<math>F</math> and <math>B</math> are collinear and <math>DF \cdot DB</math> is of constant value. Mr. Prim's blowing engine used for ventilating the House of Commons, 1877. The crosshead of the reciprocating air pump is guided by a Peaucellier linkage shown in '''Image 17'''. The slatelined air cylinders had rubberflap inlet and exhaust valves and a piston whose periphery was formed by two rows of brush bristles. Prim's machine was driven by a steam engine.'''Ferguson, 1962, p. 205''' 
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{{!}}[[Image:Blowing engine.jpgcenterborder600px]]  {{!}}[[Image:Blowing engine.jpgcenterborder600px]]  
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  {{!}}align="center"{{!}}'''Image 17'''  +  {{!}}align="center"{{!}}'''Image 17''' '''Ferguson, 1962, p. 205''' 
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=='''Hart's Linkage'''==  =='''Hart's Linkage'''==  
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  {{!}}After PeaucellierLipkin Linkage was introduced to England in 1874, Mr. Hart of Woolwich devised a new linkage that contained only four links which is the blue part as shown in '''Image 18'''. Point <math>O</math> is the inversion center with <math>OP</math> and <math>OQ</math> collinear and <math>OP \cdot OQ =</math> constant. When point <math>P</math> is constrained to move in a circle that passes through point <math>O</math>, then point <math>Q</math> will trace out a straight line. See below for proof.  +  {{!}}After PeaucellierLipkin Linkage was introduced to England in 1874, Mr. Hart of Woolwich Academy '''Kempe, 1877, p. 18''' devised a new linkage that contained only four links which is the blue part as shown in '''Image 18'''. Point <math>O</math> is the inversion center with <math>OP</math> and <math>OQ</math> collinear and <math>OP \cdot OQ =</math> constant. When point <math>P</math> is constrained to move in a circle that passes through point <math>O</math>, then point <math>Q</math> will trace out a straight line. See below for proof. 
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{{!}}[[Image:Hartlinkage3.pngbordercenter600px]]  {{!}}[[Image:Hartlinkage3.pngbordercenter600px]]  
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{{!}}[[Image:Circle in circle 1.pngbordercenter325px]]{{!}}{{!}}[[Image:Circle in circle 2.pngbordercenter300px]]{{!}}{{!}}[[Image:Img335.gifbordercenter300px]]  {{!}}[[Image:Circle in circle 1.pngbordercenter325px]]{{!}}{{!}}[[Image:Circle in circle 2.pngbordercenter300px]]{{!}}{{!}}[[Image:Img335.gifbordercenter300px]]  
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  {{!}}align="center"{{!}}'''Image 19'''{{!}}{{!}}align="center"{{!}}'''Image 20'''{{!}}{{!}}align="center"{{!}}'''Image 21'''  +  {{!}}align="center"{{!}}'''Image 19'''{{!}}{{!}}align="center"{{!}}'''Image 20'''{{!}}{{!}}align="center"{{!}}'''Image 21''' '''Bryant, & Sangwin, 2008, p.44''' 
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  {{!}}colspan="3"{{!}}There are many other mechanisms that create straight line. I will only introduce one of them here. Refer to '''Image 19'''. Consider two circles <math>C_1</math> and <math>C_2</math> with radius having the relation <math>2r_2=r_1</math>. We roll <math>C_2</math> inside <math>C_1</math> without slipping as show in '''Image 20'''. Then the arch lengths <math>r_1\beta = r_2\alpha</math>. Voila! <math>\alpha = 2\beta</math> and point <math>C</math> has to be on the line joining the original points <math>P</math> and <math>Q</math>! The same argument goes for point <math>P</math>. As a result, point <math>C</math> moves in the horizontal line and point <math>P</math> moves in the vertical line. In 1801, James White patented his mechanism using this rolling motion. It is shown in '''Image 21'''.  +  {{!}}colspan="3"{{!}}There are many other mechanisms that create straight line. I will only introduce one of them here. Refer to '''Image 19'''. Consider two circles <math>C_1</math> and <math>C_2</math> with radius having the relation <math>2r_2=r_1</math>. We roll <math>C_2</math> inside <math>C_1</math> without slipping as show in '''Image 20'''. Then the arch lengths <math>r_1\beta = r_2\alpha</math>. Voila! <math>\alpha = 2\beta</math> and point <math>C</math> has to be on the line joining the original points <math>P</math> and <math>Q</math>! The same argument goes for point <math>P</math>. As a result, point <math>C</math> moves in the horizontal line and point <math>P</math> moves in the vertical line. In 1801, James White patented his mechanism using this rolling motion. It is shown in '''Image 21''''''Bryant, & Sangwin, 2008, p.4244'''. 
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{{!}}colspan="3" align="center"{{!}}[[Image:Ellipsograph2.pngbordercenter500px]]  {{!}}colspan="3" align="center"{{!}}[[Image:Ellipsograph2.pngbordercenter500px]]  
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{{!}}colspan="3" align="center"{{!}}'''Image 22'''  {{!}}colspan="3" align="center"{{!}}'''Image 22'''  
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  {{!}}colspan="3"{{!}}Interestingly, if you attach a rod of fixed length to point <math>C</math> and <math>P</math> and the end of the rod <math>T</math> will trace out an ellipse as seen in '''Image 22'''. Why? Consider the coordinates of <math>P</math> in terms of <math>\theta</math>, <math>PT</math> and <math>CT</math>. Point <math>T</math> will have the coordinates <math>(CT \cos \theta, PT \sin \theta)</math>. Now, whenever we see <math>\cos \theta</math> and <math>\sin \theta</math> together, we want to square them. Hence, <math>x^2=CT^2 \cos^2 \theta</math> and <math>y^2=PT^2 \sin^2 \theta</math>. Well, they are not so pretty yet. So we make them pretty by dividing <math>x^2</math> by <math>CT^2</math> and <math>y^2</math> by <math>PT^2</math>, obtaining <math>\frac {x^2}{CT^2} = \cos^2 \theta</math> and <math>\frac {y^2}{PT^2} = \sin^2 \theta</math>. Voila again! <math>\frac {x^2}{CT^2} + \frac {y^2}{PT^2}=1</math> and this is exactly the algebraic formula for an ellipse.  +  {{!}}colspan="3"{{!}}Interestingly, if you attach a rod of fixed length to point <math>C</math> and <math>P</math> and the end of the rod <math>T</math> will trace out an ellipse as seen in '''Image 22'''. Why? Consider the coordinates of <math>P</math> in terms of <math>\theta</math>, <math>PT</math> and <math>CT</math>. Point <math>T</math> will have the coordinates <math>(CT \cos \theta, PT \sin \theta)</math>. Now, whenever we see <math>\cos \theta</math> and <math>\sin \theta</math> together, we want to square them. Hence, <math>x^2=CT^2 \cos^2 \theta</math> and <math>y^2=PT^2 \sin^2 \theta</math>. Well, they are not so pretty yet. So we make them pretty by dividing <math>x^2</math> by <math>CT^2</math> and <math>y^2</math> by <math>PT^2</math>, obtaining <math>\frac {x^2}{CT^2} = \cos^2 \theta</math> and <math>\frac {y^2}{PT^2} = \sin^2 \theta</math>. Voila again! <math>\frac {x^2}{CT^2} + \frac {y^2}{PT^2}=1</math> and this is exactly the algebraic formula for an ellipse. '''Cundy, & Rollett, 1961, p. 240''' 
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*)http://www.howround.com/  *)http://www.howround.com/  
References=  References=  
  Bryant, John, & Sangwin, Christopher. (2008). How Round is your circle?. Princeton & Oxford: Princeton Univ Pr. (Bryant, & Sangwin, 2008)  +  #Bryant, John, & Sangwin, Christopher. (2008). How Round is your circle?. Princeton & Oxford: Princeton Univ Pr. (Bryant, & Sangwin, 2008) 
  Cundy, H.Martyn, & Rollett, A.P. (1961). Mathematical models. Clarendon, Oxford : Oxford University Press.(Cundy, & Rollett, 1961)  +  #Cundy, H.Martyn, & Rollett, A.P. (1961). Mathematical models. Clarendon, Oxford : Oxford University Press.(Cundy, & Rollett, 1961) 
  Henderson, David. (2001). Experiencing geometry. Upper Saddle River, New Jersey: Prentice hall. (Henderson, 2001)  +  #Henderson, David. (2001). Experiencing geometry. Upper Saddle River, New Jersey: Prentice hall. (Henderson, 2001) 
  Kempe, A. B. (1877). How to Draw a straight line; a lecture on linkage. London: Macmillan and Co.. (Kempe, 1877)  +  #Kempe, A. B. (1877). How to Draw a straight line; a lecture on linkage. London: Macmillan and Co.. (Kempe, 1877) 
  Taimina, D. (n.d.). How to Draw a Straight Line. Retrieved from The Kinematic Models for Design Digital Library: http://kmoddl.library.cornell.edu/tutorials/04/ (Taimina)  +  #Taimina, D. (n.d.). How to Draw a Straight Line. Retrieved from The Kinematic Models for Design Digital Library: http://kmoddl.library.cornell.edu/tutorials/04/ (Taimina) 
  Ferguson, Eugene S. (1962). Kinematics of mechanisms from the time of watt. United States National Museum Bulletin, (228), 185230. (Ferguson, 1962)  +  #Ferguson, Eugene S. (1962). Kinematics of mechanisms from the time of watt. United States National Museum Bulletin, (228), 185230. (Ferguson, 1962) 
+  #http://en.wikipedia.org/wiki/Wikipedia:Citing_sources  
+  #Weisstein, Eric W. Great Circle. Retrieved from MathWorldA Wolfram Web Resource: http://mathworld.wolfram.com/GreatCircle.html (Weisstein)  
+  #Wikipedia (Steam Engine). (n.d.). Steam Engine. Retrieved from Wikipedia: http://en.wikipedia.org/wiki/Steam_engine  
+  #Wikipedia (Watt's Linkage). (n.d.). Watt's Linkage. Retrieved from Wikipedia: http://en.wikipedia.org/wiki/Watt%27s_linkage  
+  #Lienhard, J. H. (1999, February 18). "I SELL HERE, SIR, WHAT ALL THE WORLD DESIRES TO HAVE  POWER". Retrieved from The Engines of Our Ingenuity: http://www.uh.edu/engines/powersir.htm (Lienhard, 1999, February 18)  
+  #Wikipedia (Peaucellier–Lipkin linkage). (n.d.). Peaucellier–Lipkin linkage. Retrieved from Wikipedia: http://en.wikipedia.org/wiki/Peaucellier%E2%80%93Lipkin_linkage  
ToDo=I need to change the size of the main picture and maybe some more theoretical description what a straight line here.  ToDo=I need to change the size of the main picture and maybe some more theoretical description what a straight line here.  
InProgress=Yes  InProgress=Yes  
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Revision as of 15:44, 9 July 2010
 Independently invented by a French army officer, CharlesNicolas Peaucellier and a Lithuanian (Nationality disputable; some say he is Russian.) mathematician Lipmann Lipkin, this is the first planar linkage that drew a straight line without using a straight edge and it had important applications in engineering and mathematics.^{[1]}^{[2]}^{[3]}
How to draw a straight line without a straight edge 

Introduction
What is a straight line? How do you define straightness? How do you construct something straight without assuming you have a straight edge? These are questions that seem silly to ask because they are so intuitive. We come to accept that straightness is simply straightness and its definition, like that of point and line, is simply assumed. However, compare this to the way we draw a circle. When using a compass to draw a circle, we are not starting with a figure that we accept as circular; instead, we are using a fundamental property of circles, that the points on a circle are at a fixed distance from the center. This page explores the properties of straight lines and more importantly and interestingly their constructions.
What Is A Straight Line? A Question Rarely Asked.
Today, we simply define a line as a onedimensional object that extents to infinity in both directions and it is straight, i.e. no wiggles along its length. But what is straightness? It is a hard question because we have the picture in our head and the answer right there under our breath but we simply cannot articulate it.
 
Image 1  Image 2 Weisstein 
The Quest to Draw a Straight Line
The Practical Need
James Watt's breakthrough
James Watt found a mechanism that converted the linear motion of pistons in the cylinder to the semi circular motion of the beam (or the circular motion of the flywheel) and vice versa. In 1784, he invented a three member linkage that solved the linear motion to circular problem practically as illustrated by the animation below. In its simplest form, there are two radius arms that have the same lengths and a connecting arm with midpoint P. Point P moves in a straight line. However, this linkage only produced approximate straight line (a stretched figure 8 actually) as shown in Image 7, much to the chagrin of the mathematicians who were after absolute straight lines. There is a more general form of the Watt's linkage that the two radius arms having different lengths like shown in Image 6. To make sure that Point P still move in the stretched figure 8, it has to be positioned such that it adheres to the ratio.Bryant, & Sangwin, 2008, p. 24  
Image 6 Bryant, & Sangwin, 2008, p. 23  Image 7 Wikipedia (Watt's Linkage) 
The Motion of Point P
We intend to described the path of so that we could show it does not move in a straight line (which is obvious in the animation) and more importantly to pinpoint the position of using certain parameter we know such as the angle of rotation or one coordinate of point . This is awfully crucial in engineering as engineers would like to know that there are no two parts of the machine will collide with each other throughout the motion. In addition, you can use the parametrization to create your own animation like that in Image 7.
Algebraic Description
Parametric Description
The First Planar Straight Line Linkage  PeaucellierLipkin Linkage
Image 13 Wikipedia (Peaucellier–Lipkin linkage)  Mathematicians and engineers had being searching for almost a century to find the solution to the straight line linkage but all had failed until 1864, a French army officer Charles Nicolas Peaucellier came up with his inversor linkage. Interestingly, he did not publish his findings and proof until 1873, when Lipmann I. Lipkin, a student from University of St. Petersburg, demonstrated the same working model at the World Exhibition in Vienna. Peaucellier acknowledged Lipkin's independent findings with the publication of the details of his discovery in 1864 and the mathematical proof. Taimina

Image 14  
Let's turn to a skeleton drawing of the PeaucellierLipkin linkage in Image 14. It is constructed in such a way that and . Furthermore, all the bars are free to rotate at every joint and point is a fixed pivot. Due to the symmetrical construction of the linkage, it goes without proof that points , and lie in a straight line. Construct lines and and they meet at point .
Since shape is a rhombus and Now,
Therefore, Let's take a moment to look at the relation . Since the length and are of constant length, then the product is of constant value however you change the shape of this construction.  
Image 15  
Refer to Image 15. Let's fix the path of point such that it traces out a circle that has point on it. is the the extra link pivoted to the fixed point with . Construct line that cuts the circle at point . In addition, construct line such that .
Since,
and Therefore constant, i.e. the length of (or the xcoordinate of w.r.t ) does not change as points and move. Hence, point moves in a straight line. ∎Bryant, & Sangwin, 2008, p. 3336 
Inversive Geometry in PeaucellierLipkin Linkage
As a matter of fact, the first part of the proof given above is already sufficient. Due to inversive geometry, once we have shown that points , and are collinear and that is of constant value. Points and are inversive pairs with as inversive center. Therefore, once moves in a circle that contains , then will move in a straight line and vice versa. ∎ See Inversion for more detail.
PeaucellierLipkin Linkage in Action
Image 16 
The new linkage caused considerable excitement in London. Mr. Prim, "engineer to the House", utilized the new compact form invented by H.Hart to fit his new blowing engine which proved to be "exceptionally quiet in their operation." In this compact form, , and . Point and are fixed pivots. In Image 16. F is the inversive center and points , and are collinear and is of constant value. Mr. Prim's blowing engine used for ventilating the House of Commons, 1877. The crosshead of the reciprocating air pump is guided by a Peaucellier linkage shown in Image 17. The slatelined air cylinders had rubberflap inlet and exhaust valves and a piston whose periphery was formed by two rows of brush bristles. Prim's machine was driven by a steam engine.Ferguson, 1962, p. 205 
Image 17 Ferguson, 1962, p. 205 
Hart's Linkage
After PeaucellierLipkin Linkage was introduced to England in 1874, Mr. Hart of Woolwich Academy Kempe, 1877, p. 18 devised a new linkage that contained only four links which is the blue part as shown in Image 18. Point is the inversion center with and collinear and constant. When point is constrained to move in a circle that passes through point , then point will trace out a straight line. See below for proof. 
Image 18 
We know that
As a result, Draw line , intersecting at point . Consequently, points are collinear Construct rectangle
For , We then have . Further, due to where We have 
Other Straight Line Mechanism
Image 19  Image 20  Image 21 Bryant, & Sangwin, 2008, p.44 
There are many other mechanisms that create straight line. I will only introduce one of them here. Refer to Image 19'. Consider two circles and with radius having the relation . We roll inside without slipping as show in Image 20. Then the arch lengths . Voila! and point has to be on the line joining the original points and ! The same argument goes for point . As a result, point moves in the horizontal line and point moves in the vertical line. In 1801, James White patented his mechanism using this rolling motion. It is shown in Image 21'Bryant, & Sangwin, 2008, p.4244.  
Image 22  
Interestingly, if you attach a rod of fixed length to point and and the end of the rod will trace out an ellipse as seen in Image 22. Why? Consider the coordinates of in terms of , and . Point will have the coordinates . Now, whenever we see and together, we want to square them. Hence, and . Well, they are not so pretty yet. So we make them pretty by dividing by and by , obtaining and . Voila again! and this is exactly the algebraic formula for an ellipse. Cundy, & Rollett, 1961, p. 240 
ConclusionThe Take Home Message
We should not take the conception of straight line for granted and there are many interesting, and important, issues surrounding the concepts of straight line. A serious exploration of its properties and constructions will not only give you a glimpse of geometry's all encompassing reach into science, engineering and our lives, but also make you question many of the assumptions you have about geometry. Hopefully, you will start questioning the flatness of a plane, roundness of a circle and the nature of a point. This was how real science and amazing discoveries were made and this is how you should learn and appreciate them.
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About the Creator of this Image
KMODDL is a collection of mechanical models and related resources for teaching the principles of kinematicsthe geometry of pure motion. The core of KMODDL is the Reuleaux Collection of Mechanisms and Machines, an important collection of 19thcentury machine elements held by Cornell's Sibley School of Mechanical and Aerospace Engineering.
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Additional Resources
 )http://kmoddl.library.cornell.edu/model.php?m=244
 )http://dlxs2.library.cornell.edu/cgi/t/text/textidx?c=math;cc=math;view=toc;subview=short;idno=Kemp009
 )http://kmoddl.library.cornell.edu/tutorials/04/
 )http://www.howround.com/
References
 Bryant, John, & Sangwin, Christopher. (2008). How Round is your circle?. Princeton & Oxford: Princeton Univ Pr. (Bryant, & Sangwin, 2008)
 Cundy, H.Martyn, & Rollett, A.P. (1961). Mathematical models. Clarendon, Oxford : Oxford University Press.(Cundy, & Rollett, 1961)
 Henderson, David. (2001). Experiencing geometry. Upper Saddle River, New Jersey: Prentice hall. (Henderson, 2001)
 Kempe, A. B. (1877). How to Draw a straight line; a lecture on linkage. London: Macmillan and Co.. (Kempe, 1877)
 Taimina, D. (n.d.). How to Draw a Straight Line. Retrieved from The Kinematic Models for Design Digital Library: http://kmoddl.library.cornell.edu/tutorials/04/ (Taimina)
 Ferguson, Eugene S. (1962). Kinematics of mechanisms from the time of watt. United States National Museum Bulletin, (228), 185230. (Ferguson, 1962)
 http://en.wikipedia.org/wiki/Wikipedia:Citing_sources
 Weisstein, Eric W. Great Circle. Retrieved from MathWorldA Wolfram Web Resource: http://mathworld.wolfram.com/GreatCircle.html (Weisstein)
 Wikipedia (Steam Engine). (n.d.). Steam Engine. Retrieved from Wikipedia: http://en.wikipedia.org/wiki/Steam_engine
 Wikipedia (Watt's Linkage). (n.d.). Watt's Linkage. Retrieved from Wikipedia: http://en.wikipedia.org/wiki/Watt%27s_linkage
 Lienhard, J. H. (1999, February 18). "I SELL HERE, SIR, WHAT ALL THE WORLD DESIRES TO HAVE  POWER". Retrieved from The Engines of Our Ingenuity: http://www.uh.edu/engines/powersir.htm (Lienhard, 1999, February 18)
 Wikipedia (Peaucellier–Lipkin linkage). (n.d.). Peaucellier–Lipkin linkage. Retrieved from Wikipedia: http://en.wikipedia.org/wiki/Peaucellier%E2%80%93Lipkin_linkage
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I need to change the size of the main picture and maybe some more theoretical description what a straight line here.
Leave a message on the discussion page by clicking the 'discussion' tab at the top of this image page.
In Euclid's book Elements, he defined a straight line as "lying evenly between its extreme points" and it has "breadthless width." The definition is pretty useless. What does he mean if he says "lying evenly"? It tells us nothing about how to describe or construct a straight line. So what is a straightness anyway? There are a few good answers. For instance, in the Cartesian Coordinates, the graph of is a straight line. In addition, the shortest distance between two points on a flat plane is a straight line, a definition we are most familiar with. However, it is important to realize that the definitions of being "shortest" and "straight" will change when you are no longer on flat plane. For example, the shortest distance between two points on a sphere is the the "great circle", a section of a sphere that contains a diameter of the sphere, and great circle is straight on the spherical surface.
For more properties on straight line, you can refer to the book Experiencing Geometry by David W. Henderson.