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CURVES IN ENGINEERING

An attempt on lucidity & holism PON.RATHNAVEL

SyllabusConics – Construction of ellipse, Parabola and hyperbola by eccentricity method – Construction of cycloid and involutes of square and circle – Drawing of tangents and normal to the above curves. 10 hours

SynopsisIntroduction to Curves – Classification of Curves – Introduction to Conics, Roulettes and Involutes Terminology in Curves - Properties of Conics, Roulettes and Involutes - Construction of ellipse by eccentricity method - Construction of Parabola by eccentricity method - Construction of hyperbola by eccentricity method – Construction of cycloid – Construction of Involute of square – Construction of Involute of Circle 05 periods

WHY CURVES?

CIVIL ENGINEERINGBridges, Arches, Dams, Roads, Manholes etc.

MECHANICAL ENGINEERINGGear Teeth, Reflector Lights, Centrifugal Pumps etc

ECEDesign of Satellites, Missiles etc, Dish Antennas, ECG & EEG Machines

CSE & ITComputer Graphics, Networking Concepts

ENGINEERING GRAPHICS EXAM2 Marks - 4 & 15 Marks - 1

JUMBLE ?

U O L CS

LOCUS

SET OF POINTS

GIVEN CONDITIONS

PATH Vs LOCUS

Locus is a collection of points which share a property.

It is used to define curves in a geometry.

CURVE

A curve is considered to

be the locus of a set of points that satisfy an

algebraic equation

CLASSIFICATIONCURVES

CONIC SECTIONS ENGINEERING CURVES

1. CIRCLE2. ELLIPSE3. PARABOLA4. HYPERBOLA5. RECTANGULAR HYPERBOLA

1. CYCLOIDAL CURVES/ROULETTESa.Cycloidb.EpiCycloidc.Hypocycloidd.Trochoids(Superior & Inferior)e.Epitrochoids(Superior & Inferior)f.Hypotrochoids(Superior&Inferior)2. INVOLUTE3. SPIRALSa.Archimedianb.Logarithmicc.Hyperbolic4. HELICESa.Cylindricalb.Conical5. SPECIAL CURVES

STICKING TO SYLLABUS

Theory

CONICSROULETTESINVOLUTES

Practical

ELLIPSEPARABOLA

HYPERBOLACYCLOID

INVOLUTE OF SQUAREINVOLUTE OF CIRCLE

CONIC SECTIONS (A) CONICS

The curves obtained by the intersection of a cone by cutting plane in different positions are called conics.

The conics are1. CIRCLE2. ELLIPSE3. PARABOLA4. HYPERBOLA5. RECTANGULAR HYPERBOLA

DEFINING CONICS

Curve Position of Cutting Plane

Circle Perpendicular to axis and parallel to the base

Ellipse Inclined to the axis and not parallel to any generator. Angle of Cutting Plane > Angle of Generator

Parabola Inclined to axis, parallel to generators and passes through the base and axis

Hyperbola Inclined to the axis and not parallel to any generator. Angle of Cutting Plane < Angle of Generator

Rectangular Hyperbola

Parallel to the Axis and Perpendicular to the Base

ELLIPSE

Ellipse is defined as the locus of points the sum of whose distances from two fixed points, called the foci, is a constant.

PARABOLA

Parabola is defined as the locus of points whose distances from a fixed point, called the focus, and a fixed line, called the directrix, are always equal.

HYPERBOLA

Hyperbola is defined as the locus of points whose distances from two fixed points, called the foci, remains constant.

ROULETTES

A cycloid is a curve generated by a point on the circumference of a circle as the circle rolls along a straight line without slipping.

The rolling circle is called generating circle and the line along which it rolls is called base line or directing line.

ROULETTES

CYCLOID – THE QUARREL CURVE OR THE HELEN OF GEOMETRY

ROULETTES

CYCLOID – THE QUARREL CURVE OR THE HELEN OF GEOMETRY

ROULETTES

An epicycloid is a curve generated by a point on the circumference of a circle which rolls on the outside of another circle without sliding or slipping.

The rolling circle is called generating circle and the outside circle on which it rolls is called the directing circle or the base circle.

ROULETTES

ROULETTES

ROULETTES

A hypocycloid is a curve generated by a point on the circumference of a circle which rolls on the inside of another circle without sliding or slipping.

The rolling circle is called generating circle/hypocircle and the inside circle on which it rolls is called the directing circle or the base circle.

ROULETTES

ROULETTES

ROULETTES

A trochoid is a curve generated by a point either inside or outside the circumference of a circle that rolls along a straight line without slipping.

When the point is inside the circumference of the circle, it is called inferior trochoid. If it is outside the circumference of the circle, it is called superior trochoid.

An inferior trochoid is also called prolate cycloid.A superior trochoid is also called curtate cycloid.

ROULETTES

An epitrochoid is a curve generated by a point either inside or outside the circumference of a circle that rolls along the outside of an circle without slipping.

When the point is inside the circumference of the circle, it is called inferior epitrochoid. If it is outside the circumference of the circle, it is called superior epitrochoid.

ROULETTES

A hypotrochoid is a curve generated by a point either inside or outside the circumference of a circle that rolls along the outside of an circle without slipping.

When the point is inside the circumference of the circle, it is called inferior hypotrochoid. If it is outside the circumference of the circle, it is called superior hypotrochoid.

INVOLUTES

An involute is a curve traced by a point as it unwinds from around a circle or polygon.

The concerned circle or polygon is called as evolute.

INVOLUTES

INVOLUTES

INVOLUTES