The profile of objects consists of different types of shapes which is called as curves.
commonly used in engineering practice are: Conic sections, Cycloidal curves, Involute, Evolutes, Spirals, Helix.
The sections obtained by the intersection of a right circular cone by a plane in different positions relative to the axis of the cone are called conics. By cutting the cones, we can obtain different curves like circle, ellipse, parabola, hyperbola etc.
The conic may also be defined as the locus of a point moving in a plane in such a way that the ratio of its distances from a fixed point and a fixed straight line is always constant.The fixed point is called the focus and the fixed line, the directrix & the ratio Is called as eccentricity and is denoted by e.
When the section plane is inclined to the axis and cuts all the generators on one side of the apex, the section is an ellipse.
Eccentricity of an ellipse is always less than 1.
Method of formation of an ellipse -
(i) Arc of circles method
(ii) Concentric circles method
(iii) Loop of the thread method
(iv) Oblong method
(v)Trammel or Elliptical trammel method
When the section plane is inclined to the axis and is parallel to one of the generators, the section obtained by the cutting is a parabola. Eccentricity of a parabola is equal to 1.
Mathematically a parabola can be described by an equation y2 =4ax or x2 = 4ay.
y2 =4ax ⇒ Rightward Parabola, x2 = 4ay ⇒ Upward Parabola
A hyperbola is a plane curve having two separate parts or branches, formed when two cones that point towards one another are intersected by a plane that is parallel to the axes of the cones.
Eccentricity of a hyperbola greater than 1 for hyperbola.
It is a curve traced out by a point moving in such a way that the product of its distances from two fixed lines at right angles to each other is a constant. The fixed lines are called as Asymptotes.
Equation of rectangular hyperbola xy = Constant
These curves are generated by a fixed point on the circumference of a circle, which rolls without slipping along a fixed straight line or a circle.
Generating Circle- Circle which rolls on the line or another circle is called as generating circle
Directing Line/Circle – Line or circle in which Generating circle rolls is termed directing line or directing circle.
Cycloid: Cycloid is a curve generated by a point on the circumference of a circle which rolls along a straight line.
Mathematically it can be expressed as,
y = a (1 – cos θ) or x = a (θ – sin θ)
Note- The normal at any point on a cycloidal curve will pass through the corresponding point of contact between the generating circle and the directing line or circle.
Trochoid: Trochoid is a curve generated by a point fixed to a circle, within or outside its circumference, as the circle rolls along a straight line.
(i) Inferior Trochoid: When the point is within the circle, the curve is called as inferior trochoid.
(ii) Superior Trochoid: When the point is outside the circle, the curve is called as Superior trochoid
(iii) Epitrochoid: Epitrochoid is a curve generated by a point fixed to a circle (within or outside its circumference, but in the same plane) rolling on the outside of another circle.
(iv) Hypotrochoid: When the circle rolls inside another circle, the curve is called a hypotrochoid.
Epicycloid: The curve generated by a point on the circumference of a circle, which rolls outside of another circle without slipping, is called an epicycloid.
Hypocycloid: The curve generated by a point on the circumference of a circle, which rolls inside of another circle without slipping, is called an epicycloid.
Note: When the diameter of the rolling circle is half the diameter of the directing circle, the hypocycloid is a straight line and is a diameter of the directing circle.
The involute is a curve traced out by an end of a piece of thread unwound from a circle or a polygon, the thread being kept tight.
Thread will always be tangent to the circle thus normal to an involute of a circle is tangent to that circle.
It may also be defined as a curve traced out by a point in a straight line which rolls without slipping along a circle or a polygon.
Involute of a circle is used as teeth profile of gear wheel.
Involute of a square:
If a line rotates in a plane about one of its ends and if at the same time, a point moves along the line continuously in one direction, the curve traced out by the moving point is called a spiral.
The point about which the line rotates is called a pole.
The line joining any point on the curve with the pole is called the radius vector and the angle between this line and the line in its initial position is called the vectorial angle.
Each complete revolution of the curve is termed the convolution.
A spiral may make any number of convolutions before reaching the pole.
(i) Archemedian Spiral:
It is a curve traced out by a point moving in such a way that its movement towards or away from the pole is uniform with the increase of the vectorial angle from the starting line.
The use of this curve is made in teeth profiles of helical gears, profiles of cams etc.
(ii) Logarithmic or Equiangular spiral:
In a logarithmic spiral, the ratio of the lengths of consecutive radius vectors enclosing equal angles is always constant.
In other words, the values of vectorial angles are in arithmetical progression and the corresponding values of radius vectors are in geometrical progression.
Helix is defined as a curve, generated by a point, moving around the surface of a right circular cylinder or a right circular cone in such a way that, its axial movement in the direction of the axis of the cylinder or the cone is uniform with its movement around the surface of the cylinder or the cone. The axial advance of the point during one complete revolution is called the pitch of the helix.
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