Engineering Curves: Conic Sections, Cycloids & Involutes

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1. Conic Sections: Fundamental Definitions

A conic section is the locus of a point $P$ that moves in a plane such that the ratio of its distance from a fixed point (Focus, $F$) to its perpendicular distance from a fixed straight line (Directrix, $DD'$) is a constant called Eccentricity ($e$).

Directrix (DD')
    |
    |          P (Moving point)
    |          /|
  M +---------+ | 
    |         \ |
    |          \|
----+-----------F----------- Axis (perpendicular to directrix)
  C |           (Focus)
    |

Conic Curve	Eccentricity ($e$)	Characteristic Shape	Engineering Applications
Ellipse	$e < 1$ (e.g., $2/3$)	Closed, symmetric oval	Planetary orbits, elliptical gears, bridges, arches.
Parabola	$e = 1$	Open curve, arms go to $\infty$	Reflectors, headlamps, suspension bridge cables, solar troughs.
Hyperbola	$e > 1$ (e.g., $3/2$)	Open dual-branched curve	Cooling towers, supersonic shockwaves, navigation (LORAN).
Rectangular Hyperbola	$e = \sqrt{2}$, asymptotes $\perp$	$PV = \text{constant}$	Boyle's Law ($P\text{-}V$ indicator diagrams), heat exchangers.

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2. General (Eccentricity / Focus-Directrix) Method

The General Method constructs any conic directly from its directrix $DD'$, focus $F$, and eccentricity $e$.

Step-by-Step Procedure:

1. Axis & Directrix: Draw vertical line $DD'$ (Directrix). Draw horizontal line $CF$ perpendicular to $DD'$ through point $C$ (Axis of the conic).
2. Locate Focus $F$: Mark focus $F$ on the axis at the given distance from $C$ (e.g., $CF = 50\text{ mm}$).
3. Locate Vertex $V$: Divide distance $CF$ into $(m + n)$ equal parts, where $e = m/n$:
   • For Ellipse ($e = 2/3$): Divide $CF$ into $2 + 3 = 5$ parts. Vertex $V$ is at $2$ parts from $F$ and $3$ parts from $C$ ($VF/VC = 2/3$).
   • For Parabola ($e = 1$): Divide $CF$ into $2$ equal parts ($VF = VC$).
   • For Hyperbola ($e = 3/2$): Divide $CF$ into $3 + 2 = 5$ parts. Vertex $V$ is at $3$ parts from $F$ and $2$ parts from $C$ ($VF/VC = 3/2$).
4. Plotting Points:
   • Mark several vertical ordinates $1, 2, 3, 4, \dots$ along the axis to the right of $V$.
   • Erect vertical lines through $1, 2, 3, \dots$.
   • Measure distance from directrix $DD'$ to point $1$ (distance $C1$).
   • Set compass to radius $R_1 = e \times C1$ (for parabola, $R_1 = C1$).
   • With compass centered at Focus $F$, strike arcs cutting the vertical ordinate at point $1$ above and below the axis ($P_1, P_1'$).
   • Repeat for points $2, 3, 4, \dots$.
5. Draw the Curve: Join points $V, P_1, P_2, \dots$ with a smooth French curve.

Tangent and Normal Construction:

To draw a tangent and normal at any point $P$ on the conic:

1. Join point $P$ to the Focus $F$.
2. At focus $F$, construct line $FT$ perpendicular to line $PF$ ($\angle PFT = 90^\circ$), cutting the directrix at point $T$.
3. Join $T$ and $P$. Line $TP$ is the Tangent.
4. Draw line $NP$ perpendicular to $TP$ at $P$. Line $NP$ is the Normal.

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3. Cycloidal Curves

A cycloid is the locus of a point on the circumference of a generating circle rolling along a fixed path without slipping.

• Cycloid: Generating circle rolls on a straight baseline (base length $= \pi D$).
• Epicycloid: Generating circle rolls on the outside of a directing circle of radius $R$.
• Hypocycloid: Generating circle rolls on the inside of a directing circle of radius $R$.

         Generating Circle
              (O)
             /   \
            |  P  |
             \   /
============== x ===================== Baseline (L = pi * D)

Construction of a Cycloid:

1. Draw generating circle of diameter $D$. Draw horizontal baseline tangent to circle at bottom, of length $L = \pi D$.
2. Divide circle into $12$ equal parts ($1, 2, \dots, 12$).
3. Divide baseline $L$ into $12$ equal parts ($1', 2', \dots, 12'$).
4. Draw the centerline parallel to the baseline; project points $1'$ to $12'$ to locate centers $C_1, C_2, \dots, C_{12}$.
5. Through circle divisions, draw horizontal lines parallel to baseline.
6. With center $C_1$ and radius $R = D/2$, strike an arc on horizontal line from point $1$ to get $P_1$. Repeat for $C_2 \dots C_{12}$.
7. Tangent & Normal to Cycloid:
   • For a point $P$, draw horizontal line through $P$ to center line to find center $C_k$.
   • Drop perpendicular from $C_k$ to baseline to locate instantaneous center $I$.
   • Join $IP$. $IP$ is the Normal. Line perpendicular to $IP$ at $P$ is the Tangent.

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4. Involute Curves

An involute is the path traced out by the end of an inextensible taut string unwound from a given curve (cylinder, polygon, or circle).

A. Involute of a Circle (Gear Tooth Profile):

1. Draw base circle of diameter $D$.
2. At the lowest point $P$, draw a horizontal tangent line of length equal to circumference $L = \pi D$.
3. Divide circle and tangent line into 12 equal parts ($1, 2, \dots, 12$).
4. Draw tangents to the base circle at points $1, 2, \dots, 12$ in the direction of unwinding.
5. On the tangent through $1$, mark length $P1'$ ($= 1/12 \times \pi D$). On tangent through $2$, mark length $P2'$ ($= 2/12 \times \pi D$), and so on.
6. Join points $P_1, P_2, \dots, P_{12}$ with a smooth curve.

Crucial Involute Property:

• The normal to an involute of a circle at any point is always tangent to the base circle.
• This geometric property ensures a constant pressure angle and uniform velocity ratio in involute gear teeth.

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5. Summary of Key Exam Points

1. Eccentricity Summary:
   • Circle: $e = 0$
   • Ellipse: $0 < e < 1$
   • Parabola: $e = 1$
   • Hyperbola: $e > 1$
   • Rectangular Hyperbola: $e = \sqrt{2}$
2. Tangent/Normal Rule: $\angle PFT = 90^\circ$ at the focus is universal for all conic sections.
3. Cycloid Roll Length: Exactly one revolution $= \pi D$.
4. Involute Gear Standard: Gear teeth are designed with involute profiles because their normal line always coincides with the common line of action (fixed pressure angle).