LOCI OF POINTS AND LOCI OF POINTS AND STRAIGHT LINESSTRAIGHT LINES

LOCUSLOCUS The path of a point which moves according to The path of a point which moves according to

mathematically defined conditions is known as mathematically defined conditions is known as its Locus. its Locus.

The plural of word Locus is Loci.The plural of word Locus is Loci. For example, a point P moving in a plane, so For example, a point P moving in a plane, so

that it is always at a constant distance from that it is always at a constant distance from another fixed point O traces out a circle as its another fixed point O traces out a circle as its locus. locus.

Many important geometrical curves (ellipse, Many important geometrical curves (ellipse, parabola, hyperbola, cycloidal curves) may be parabola, hyperbola, cycloidal curves) may be considered as Loci e.g, conic curves, helices, considered as Loci e.g, conic curves, helices, and screw threads, involutes and spiral curves.and screw threads, involutes and spiral curves.

Locus of point on a mechanismLocus of point on a mechanism

The locus of a point in a mechanism is the path which is The locus of a point in a mechanism is the path which is traced by the point when the mechanism moves through traced by the point when the mechanism moves through a complete cycle of operation. a complete cycle of operation.

The method of drawing the locus of a particular point in a The method of drawing the locus of a particular point in a mechanism is to construct the mechanism in several mechanism is to construct the mechanism in several positions. positions.

The point is potted for each position and its locus is The point is potted for each position and its locus is obtained by drawing a smooth curve through these obtained by drawing a smooth curve through these plotted points.plotted points.

The mechanism in successive positions may be drawn The mechanism in successive positions may be drawn with drawing instruments geometrically or with a paper with drawing instruments geometrically or with a paper trammel. The use of computer aided drafting renders the trammel. The use of computer aided drafting renders the procedure very handy and fast. procedure very handy and fast.

CYCLOIDAL AND SPIRAL CURVESCYCLOIDAL AND SPIRAL CURVES

ROULETTESROULETTES Those curves which are generated by a fixed Those curves which are generated by a fixed

point on a rolling curve that rolls without slipping point on a rolling curve that rolls without slipping along fixed base curve. The rolling curve is along fixed base curve. The rolling curve is called generating curve and the fixed curve is called generating curve and the fixed curve is called the directing curve. called the directing curve.

Some important roulettes are cycloid, epicycloid, Some important roulettes are cycloid, epicycloid, hypocycloid, trochoids, and involute. hypocycloid, trochoids, and involute.

APPLICATION OF ROULETTESAPPLICATION OF ROULETTES

CYCLOIDCYCLOID

The curve is the locus of a point on the The curve is the locus of a point on the circumference of a circle which rolls, circumference of a circle which rolls, without slipping, along a fixed straight line.without slipping, along a fixed straight line.

PROBLEMPROBLEMDraw a cycloid, given the diameter of a Draw a cycloid, given the diameter of a

generating circle as 50 mm. also draw a generating circle as 50 mm. also draw a tangent and normal at any given point T tangent and normal at any given point T on the curve. on the curve.

Solution - CycloidSolution - Cycloid

Solution - StepsSolution - Steps

With center CWith center Coo draw the rolling circle of 50 mm. draw a draw the rolling circle of 50 mm. draw a straight line, the path along which it is to roll, tangent to straight line, the path along which it is to roll, tangent to the circle.the circle.

Fix the initial position of point which is to trace the Fix the initial position of point which is to trace the required locus while the rolling circle make sone required locus while the rolling circle make sone revolution along the base line. Let it be Prevolution along the base line. Let it be Po. o.

Mark a length PMark a length Po o PPo o equal to the circuferece of the rolling equal to the circuferece of the rolling circle, along the base line, and divide it into a number of circle, along the base line, and divide it into a number of equal parts, 12 here. Divide the circumference of the equal parts, 12 here. Divide the circumference of the rolling circle also into the same number of equal parts. rolling circle also into the same number of equal parts.

Solution - StepsSolution - Steps Through division points on the rolling circle, draw lines Through division points on the rolling circle, draw lines

parallel to fixed line and at the points on the fixed line parallel to fixed line and at the points on the fixed line erect perpendiculars to cut the horizontal center line of erect perpendiculars to cut the horizontal center line of the rolling circle at points Cthe rolling circle at points C11, C, C22, C, C33 etc. etc.

As the circle rolls through 1/12As the circle rolls through 1/12thth of a complete revolution, of a complete revolution, the center Co will move to the position C1 and the point the center Co will move to the position C1 and the point P will move from initial position Po to P1 and so on. P will move from initial position Po to P1 and so on. Therefore, the points Po, P2, P3 etc. are plotted by the Therefore, the points Po, P2, P3 etc. are plotted by the intersection of lines drawn division points 1, 2, 3 etc on intersection of lines drawn division points 1, 2, 3 etc on the circle and the corresponding circle arcs drawn with the circle and the corresponding circle arcs drawn with centers C1, C2 etc, as illustrated for P4 and P5.centers C1, C2 etc, as illustrated for P4 and P5.

A smooth curve joining all the 12 points plotted thus, A smooth curve joining all the 12 points plotted thus, gives the required cycloid. gives the required cycloid.

Solution - StepsSolution - Steps

Tangent and normal at a point on the cycloidTangent and normal at a point on the cycloidDraw the rolling circle in such a position Draw the rolling circle in such a position

that It passes through T, by chain line. The that It passes through T, by chain line. The normal is given by the line TN, where N is normal is given by the line TN, where N is the point of contact between the rolling the point of contact between the rolling circle, and the fixed line. The tangent T1, circle, and the fixed line. The tangent T1, T2 is perpendicular to TN at T.T2 is perpendicular to TN at T.

TrochoidsTrochoids

When a circle rolls, without slipping along a fixed When a circle rolls, without slipping along a fixed straight line, the locus of the fixed point P not straight line, the locus of the fixed point P not lying on the rolling circle is a trochoid.lying on the rolling circle is a trochoid.

When the point P which traces the locus is When the point P which traces the locus is outside the rolling circle, the locus produced is outside the rolling circle, the locus produced is superior trochoid.superior trochoid.

When the point P is inside the rolling circle the When the point P is inside the rolling circle the locus is inferior trochoid.locus is inferior trochoid.

The construction of both trochoids is very similar The construction of both trochoids is very similar to that used for cycloid.to that used for cycloid.

ProblemProblem

Draw trochoids, given the diameter of the Draw trochoids, given the diameter of the rolling circle as 40 mm and the fixed point rolling circle as 40 mm and the fixed point P, tracing the locus, is 8 mm away from P, tracing the locus, is 8 mm away from the rolling circle.the rolling circle.

Solution - Superior TrochoidSolution - Superior Trochoid

Solution - Inferior TrochoidSolution - Inferior Trochoid

Solution Solution

The construction of both trochoids is very The construction of both trochoids is very similar to that used for cycloid. It should be similar to that used for cycloid. It should be noted however, that in each case the noted however, that in each case the circumference of the rolling circle is laid circumference of the rolling circle is laid out along the fixed line and divided into 12 out along the fixed line and divided into 12 equal parts, and the circle through the equal parts, and the circle through the given point P is divided into 12 equal given point P is divided into 12 equal parts, not the reverse. parts, not the reverse.

EpicycloidEpicycloid

When a circle rolls, without slipping, When a circle rolls, without slipping, around the outside of a fixed circle, the around the outside of a fixed circle, the locus of a point on the circumference of locus of a point on the circumference of the rolling circle is called the epicycloid.the rolling circle is called the epicycloid.

The rolling circle is called generating circle The rolling circle is called generating circle and the fixed circle is called the directing and the fixed circle is called the directing circle.circle.

ProblemProblem

Draw an epicycloid, given the radii of Draw an epicycloid, given the radii of rolling and directing circles as r = 30 mm rolling and directing circles as r = 30 mm and R = 120 mm, respectively. Also draw and R = 120 mm, respectively. Also draw a normal and a tangent at any point Q on a normal and a tangent at any point Q on the curve. the curve.

Solution - EpicycloidSolution - Epicycloid

InvoluteInvolute

When a straight line rolls, without slipping, When a straight line rolls, without slipping, on a curve, the locus of any point on the on a curve, the locus of any point on the straight line is an involute to the curve. straight line is an involute to the curve.

The involute to a circle is the locus of the The involute to a circle is the locus of the end of a taut string as it is unwound from end of a taut string as it is unwound from the surface of a cylinder or base circle.the surface of a cylinder or base circle.

ProblemProblem

Draw an involute to a circle of 50 mm. Also Draw an involute to a circle of 50 mm. Also draw a tangent and normal to it, at any draw a tangent and normal to it, at any given point on it. given point on it.

Solution – Involute to a circleSolution – Involute to a circle

ProblemProblem

Draw the involute of a circular arc which Draw the involute of a circular arc which subtends an angle (90 degrees here) at subtends an angle (90 degrees here) at the center of the circle of 120 mm. the center of the circle of 120 mm.

Involute to a circular arcInvolute to a circular arc

ProblemProblem

Draw an involute to an equilateral traingel Draw an involute to an equilateral traingel of 20 mm side. of 20 mm side.

Involute of a triangleInvolute of a triangle

Archimedean SpiralArchimedean Spiral

It is the locus of a point P which moves at It is the locus of a point P which moves at a steady rate along a line, while the line a steady rate along a line, while the line rotates at uniform speed about center, O , rotates at uniform speed about center, O , such that for each angular displacement of such that for each angular displacement of the line, the linear displacement of the the line, the linear displacement of the point is constant. point is constant.

ProblemProblem

Construct an Archimedean spiral of two Construct an Archimedean spiral of two convolutions, given the greatest and the convolutions, given the greatest and the shortest radii as 84 mm and 12 mm, shortest radii as 84 mm and 12 mm, respectively.respectively.

Archimedean Spiral ( Two Convolutions )Archimedean Spiral ( Two Convolutions )

ProblemProblem

Construct an Archimedean spiral of one Construct an Archimedean spiral of one convolution , given the radial movement of convolution , given the radial movement of the point P during one convolution as the point P during one convolution as 60mm and the initial position of P as pole 60mm and the initial position of P as pole O. O.