Engineering Curves

Conics:

It is defined as;

Conics are the curves formed by the intersection of a plane with a right circular cone.

 

Ellipse:

It is created when a plane passes through a right circular cone at an angle to the axis that is greater than the angle between the axis and the sides.

Parabola:

It is created when a plane intersects a right circular cone parallel to the side of the cone.

Hyperbola:

It is created when a plane passes through a right circular cone at an angle to the axis that is smaller than do the elements.

Cycloid:

It is a locus of a point on the periphery of a circle which rolls on a straight line path.

Involute:

It is a locus of a free end of a string when it is wound round on a circular pole.

Epi-Cycloid:

The path traced out by a point on the edge of a circle of radius rolling on the outside of a circle of radius.

Hypo-Cycloid:

The curve produced by fixed point on the circumference of a small circle of radius rolling around the inside of a large circle of radius.

Construction Of Conics

1.  Construction Of Ellipse (Concentric Circle Method):

1. Draw both axes and named them AB and CD.
2. With the intersection point, draw two concentric circles let say of 100 mm and 70 mm.
3. Divide both circle into 8 parts.
4. From points on the outer circle, draw vertical line towards line AB.
5. From points on the inner circle, draw horizontal line away from line CD.
6. Mark intersection points of vertical and horizontal lines.
7. Draw lines joining these intersected points.

2.  Construction Of Parabola (Directrix-Focus Method):

1. Locate center line perpendicular to line AB containing vertex and focus.
2. Draw line OF and bisect it and locating vertex of Parabola.
3. Locate points 1,2,3,4 at a distance of 5 mm. 
4. With these points, draw lines parallel to line AB.
5. With focus F and distance O1, draw an arc intersecting line 1.
6. Similarly with focus F and distance O2, O3, O4, draw arc intersecting lines 2, 3, 4 respectively.
7. Draw shape by joining these intersected parts.

3.  Construction Of Hyperbola (Vertices-Focus Method):

1. Draw the Transverse and focii F and F'. 
2. Locate points 1,2,3 at any particular distance.
3. With focus F and F' with distance A1, draw four arcs above and below on the either side of axes.
4. With focus F and F' with distance B1 draw four arcs intersecting the previous arcs.
5. Similarly, repeat the procedure for point 2 and 3.
6. Draw curves joining these intersected points.
7. To draw the Asymptotes, draw a circle with radius OF or OF' with center of axes or origin.
8. Draw lines by joining opposite intersected points X.

4.  Construction Of Cycloid:

1. From center C, draw a horizontal line equal to 𝜋D distance.
2. Divide this line into 8 equal parts. 
3. Divide circle into 8 parts and start numbering clockwise.
4. From these points draw horizontal lines parallel to the line PP8.
5. With radius CP, mark points P1, P2, P3, P4, P5, P6, P7 and P8.

5.  Construction Of Involute Of A Circle:

1. From center C, draw a horizontal line equal to 𝜋D distance.
2. Divide this line into 8 equal parts. 
3. Divide circle into 8 parts and start numbering clockwise.
4. With point 1 and distance P1, mark point P1.
5. Similarly, mark point P2, P3, P4, P5, P6, P7 and P8.

6. Construction Of Epi-Cycloid:

1. Calculating angle θ by putting values in the given formula in the image.
2. Divide angle θ into 8 equal parts.
3. Draw line OP and extends it.
4. With center O, locate center point C by adding both radius (r+R).
5. Divide circle into 8 equal parts.
6. Draw arc having radius O4, O3, O2, OC, O1. 
7. With point C1 with distance CP, locate blue points.
8. Similarly, locate blue points from C2, C3, C4, C5, C6, C7, C8.

7. Construction Of Hypo-Cycloid:

1. Calculating angle θ by putting values in the given formula in the image.
2. Divide angle θ into 8 equal parts.
3. Draw line OP.
4. With center O, locate center point C by subtracting both radius (R-r).
5. Divide circle into 8 equal parts.
6. Draw arc having radius O4, O3, O2, OC, O1. 
7. With point C1 with distance CP, locate green points.
8. Similarly, locate green points from C2, C3, C4, C5, C6, C7, C8.

References:

• Videos from youtube.
• Images from google images.
• GIF: https://www.mathwarehouse.com/animated-gifs/conic-sections.php
• Materials from my university notes, Engineering Drawing Textbook by N.D. Bhatt.