Conics Constructions:

• Ellipse• Parabola• Hyperbola• Helix on a cone• Helix on a cylinder• Involutes• Cycloid• Trochoid

• Epicycloid • Epitrochoid• Hypo Cycloid• Hypo Trochoid• Archemedian Spirals

EllipseEllipse is the locus of a point moving in a plane such that

the raito of its distances from the focus and the directrix is always less than 1

Methods Of construction:• Directrix-Focus method• Arcs Of Circles Method• Concentric Circles Method• Oblong Method

Eccentricity e<1

Directrix-Focus Method• Given:

-Distance between Directrix and focus, DF-Eccentricity Ratio,e

Vertex Point= Moving point, VFocus= Fixed Point, FDirectrix= Fixed Line, D

VF+VD= Given Distance

VF/VD= Eccentricity Ratio,e

Draw an ellipse using directrix focus method when the eccentricity ratio is 2/3, and the between directrix and focus is 60mm.

Draw an ellipse using directrix focus method when the eccentricity ratio is 3/4, and the between directrix and focus is 60mm. And also draw a normal and tangent to the ellipse at 40 mm from directrix.

Procedure:-Draw vertical directrix, Mark a point on it-Pass focus axis through that point,-Take the given distance on it.-Mark the vertex point & Focus as per the

eccentricity ratio,e-At vertex point draw a perpendicular.-Cut the perpendicular with an arc of radius VF,

center V.-draw an slant line through directrix point and arc

point-Take 5 to 6 Points “At any distance” between VF.-Extend lines through these points to meet slant

line, mark as ‘’.-With focus as center and distance equal to 1-1’ cut

1-1’ and so on….. - with the focus 2 as center repeat the same

procedure for the other half of the ellipse.

Drawing Tangent and Normal to the Ellipse.• Take any point on the curve at some suitable distance, • Join it with focus point• At focus take 90o angle on the constructed line.• The normal will intersect directrix.• From the intersection point draw a line passing through

the point on the curve, this line is “tangent” to the ellipse• At the point on the ellipse draw normal to the tangent,

this line represents the normal tl the ellipse

Arcs Of Circles MethodGiven:-

Case-1- Major Axis, Minor axis- Distance Between Two Foci= Major Axis- Mark Foci from end points of “Minor Axis”

Case-2- Foci Distance and Minor Axis- Mark foci’s with distance between minor

axis end points and center point mark “Major Axis”

Case-1

• The major axis of an ellipse is 120mm long and the minor axis is 80mm long. Find the foci distances and draw the ellipse by arcs of circles method. And draw a tangent to the ellipse at a point on it 30mm above the major axis.

Procedure:

• Between center and focus take points at equal distances.

• With distances between major axis ends and points marked draw arcs with foci as center.

• These arcs intersect to give the locus traced by moving point of ellipse.

• Draw Major and Minor axis and Foci points• With the Center point and radius= ½ Major and

½ Minor draw two circles.• Divide the circles into equal number of divisions.• Drop vertical projections through Major circle • Drop horizontal projections through Minor circle• The projections cut each other at points• pass the Ellipse through these points using

French curves.

Concentric Circles Method

Oblong Method • Given:

- Major Axis- Minor Axis

Procedure:- Draw rectangle passing these axis ends

withLength= Major AxisWidth= Minor Axis

Ellipse by oblong method

Construct an ellipse in a rectangle of 120 x 80mm size.

Construct an ellipse in a parallelogram of 125x90mm size having the included angle between the faces as 100 & 80o

Construction (Oblong Method)

• Divide each half side into equal number of divisions

• Join Vertical points with Upper end of Minor axis• Join Horizontal Points with lower end of minor

axis• These lines cut each other which give 1/4th part

of ellipse• Repeat the procedure “OR” use projection

method to construct the remaining part.

ParabolaParabola is the locus of a point moving in a plane such that

the ratio of its distances from the focus and the directrix is always Equal to 1

(e=1)• Methods Of construction:

-Directrix Focus Method-Rectangle Method-Tangent Method

Directrix Focus Method-Draw a vertical directrix and mark one point on it.-Draw Axis Through that point-Mark focus point at the given distance.-Plot vertex point ‘V’ pass normal through it.-Cut the normal with an arc through center ‘v’ and radius VF-Draw slant line through obtained point passing the axis point-Take 4 to 6 points between VF at any distance.-Pass perpendiculars through them to cut slant line.-With distances between marked points and points cutting the

slant line with center F cut the normals.-Pass “Parabola” using french curve through these new points.

Rectangle Method

• Given:

- Length of parabola (Base)- Height of parabola (Axis)

Procedure• Draw A line Equal to given Length• Locate mid point of the line• Draw Axis As normal through that point• Complete the rectangle with these dimensions• Divide the base and height in equal number of

points.• Join upper point of axis with vertical points.• Project lines through base points as lines to

meet the slant lines.• Pass the parabola through the obtained points.

Tangent Method• Draw Base of the parabola, mark it’s midpoint• Through midpoint pass normal equal to height.• Join end points of base with axis ends.• Divide slant lines in equal number of points.• Join the 1st point of one slant line with the end

point of other slant line.• The blank portion inhibits parabola.• Draw a smooth curve as tangent to these lines.• This curve is the required parabola.

Hyper-BolaIt is the locus of the points in a plane such

that the ratio of its distances from fixed point and the directrix is always grater than 1.

e>1Methods of construction:-Directrix-Focus Method-Rectangle Method

Directrix-Focus Method• Draw vertical directrix, mark axis point.• Pass axis equal to given length.• Mark vertex point such that, e>1.• Pass normal through vertex point.• With distance VF, center V cut the normal.• Join the axis point and this point.• Take equidistant point between VF.• Pass normal to cut slant line through these points• With center F and distance between axis point

and slant line points cut the normals.• Pass “Hyperbola” through these points.

The distance between a fixed line and fixed point is 70mm. Draw a suitable curve when the ratio fo distances between moving point and fixed point and fixed line remains as 1.

With the eccentricity ratio as one construct a curve if the distance between focus and directrix is 60mm

Rectangular Hyperbola

• Take two asymptotes, take offset distances as given• Divide horizontal axis in 5 to 6 points at any distances• Divide the vertical axis in 5 to 6 points at any distance.• Join these points with asymptote origin.• The lines will cot horizontal and vertical axes.• Through horizontal axis pass vertical projections• Through Vertical axis pass horizontal projections• These projections meet each other, through these newly

obtained points trace a smooth “Hyperbola”

Helix on a Cylinder

It is the curve generated by a point moving around the surface of the cylinder.

Procedure:• Draw plan and elevation of cylinder with given dimensions• Draw development of surface of cylinder.• Divide the cylinder plan in 12 equal parts.• Project these points in Elevation.• Join the corners of the developed view.• Divide it in 12 equal parts.• Draw vertical normal through the divided points.• These lines will intersect the diagonal.• Project intersection points in elevation.• In elevation pass helix through the obtained points.• This is the required “Helix on cylinders”

Involutes

• It is a curve traced by an open end of a piece of a thread unwound from a circle or polygon

Or

It is a curve traced out by a point in a straight line which rolls without slipping along a circle or a polygon

Involutes of a Circle

• Draw a circle, divide it into 12 parts.• Draw a line, Length= Perimeter of circle,πd• Divide the line into 12 equal parts.• Draw tangent through each division of circle.• With 1st point as center and radius equal to 11

parts on line draw arc to cut 1st tangent.• With 2nd point as center and radius equal to 10

parts on line draw arc to cut 2nd tangent.• Pass a smooth involute through the obtained

points.

Cycloids

• Cycloids are the curves generated by a fixed point on the circumference of a circle, which rolls without slipping along a fixed straight line or a circle.

The rolling circle is called Generating circle and the fixed straight line or circle is called as Directing line or Directing Circle

Cycloids

Cycloid/ trochoid

Long Cycloid/ superior trochoid

Short cycloid/ inferior trochoid

Cycloid Construction• Draw a circle equal to radius of generating circle• Draw a horizontal tangent to circle, Length=πd• Divide the line & circle into 12 equal parts• Draw a line parallel to tangent through center of circle.• Extend the divisions on tangent to center line• Draw horizontal projectors through divided circle.• With 1st point on center line & radius= radius of circle cut

the projector through 1st division of circle• Obtain the 1st displacement of point• With 2nd center and cutting the 2nd projector obtain the

next point.• With the new points pass a smooth cycloid

Epicycloid• Given

Rolling Circle Diameter, d

Directing Circle diameter, D

Θ = πd R

Θ= Inscribed angle of Epicycloid

Calculate

EpiCycloid• Take angle θ at a point, point of origin. • With the same point draw an arc of radius=

radius of generating circle.• On one of the segments draw rolling circle

touching the arc.• Divide the arc and circle into 12 equal parts.• Through each division and center point draw arcs

with origin as center.• Take 12 centers on center line with origin as fixed

point.• With each center and radius = radius of rolling

circle, cut each arc.• With the newly obtained points, inscribe a

“Epicycloid”