A scale of 1; would be } in. to the foot, or } in. to the yard. Draw a line AB, and set off distances of } in., showing feet. Draw the vertical A3, and set off 12 equal parts. Rule parallels, and complete the scale. To obtain a required distance, say 2 yds. 2 ft. 7 in., place one point of the dividers where vertical 2 meets horizontal 7, on line B2, and extend the other point to the spot where the diagonal 2 intersects horizontal 7. In making a drawing to scale, first make the scale and then take all measurements from it.
Curves such as the ellipse are important in mechanical drawings. The Ellipse is formed by an oblique plane passing through both sides of a cone (Fig. 13a). Its longest diameter is called Transverse or Major Axis, while its shortest diameter is Conjugate or Minor Axis. Any line passing through the center and terminating in the curve is a diameter. There are two points on the transverse axis equidistant from the centre, each of which is a Focus. If any point in the curve is joined by two lines to the foci, these two lines together equal the transverse axis.
<Callout type="important" title="Key Ellipse Properties">The longest diameter of an ellipse is called the Major Axis and the shortest is the Minor Axis.</Callout> The Parabola is formed when a plane cuts the cone parallel to its side (Fig. 13b). The Hyperbola forms when the cutting plane makes a greater angle with the base than the side of the cone makes (Fig. 18c).
Problem 21: To describe an ellipse, given transverse and conjugate diameters. a) By thread and pins: Let AB and CD be the two diameters. Bisect them at E and F respectively. Take OE and OF each equal to half CD. With radius AO and centre E describe arcs cutting AB in G and H. These points are foci. Tie a piece of thread round these pins, keeping it tightly stretched as you move the pencil point around.
b) By intersecting lines: Place diameters at right angles, draw parallels forming a rectangle, divide AE into equal parts, set off on AF, BG, and BH, join points with C and D. Draw lines from D through 1, 2, 3 to meet Cl, C2, C3. In the same manner, draw lines from C through the same points to meet D1, D2, D3. Through these intersections draw the curve.
c) By two circles: From centre O describe circles with diameters equal to major and minor axis respectively of the ellipse. Draw 12 radii at 30° intervals cutting both circumferences. Horizontal lines from each radius where it cuts the circumference of the smaller circle, vertical lines from points of intersection of the large circle’s circumference with the radii will intersect forming an ellipse.
d) By paper trammel: Set up axes AB and CD as before. Take a piece of paper (or ruler), make EF equal to AO, EG equal to CO. Place trammel so that G is on transverse diameter and F on conjugate diameter. E will be a point on the curve. Shift paper keeping G on transverse and F on conjugate diameters to obtain more points.
Key Takeaways
- Understanding geometric curves such as ellipses, parabolas, and hyperbolas is crucial for construction and engineering.
- Ellipse properties include the Major Axis (longest diameter) and Minor Axis (shortest diameter).
- Various methods exist to construct an ellipse including thread and pins, intersecting lines, two circles, and a paper trammel.
Practical Tips
- Use the thread and pin method for constructing ellipses when precision is critical.
- The intersecting lines technique can be used effectively in field conditions with minimal equipment.
- For large-scale projects, consider using the paper trammel or two circles methods to ensure accuracy.
Warnings & Risks
- Improper construction of geometric curves can lead to structural weaknesses and failures.
- Always verify measurements before finalizing designs to avoid costly mistakes.
Modern Application
While this chapter focuses on historical techniques for constructing geometric shapes, these principles remain foundational in modern engineering and survival scenarios. Understanding how to accurately draw ellipses and other conic sections can be invaluable when designing structures or tools under primitive conditions.
Frequently Asked Questions
Q: What are the key properties of an ellipse mentioned in this chapter?
The longest diameter of an ellipse is called the Major Axis, while the shortest diameter is referred to as the Minor Axis.
Q: How can one construct an ellipse using a paper trammel?
To use a paper trammel for constructing an ellipse, set up axes AB and CD. Take a piece of paper or ruler, make EF equal to AO, and EG equal to CO. Place the trammel so that G is on the transverse diameter and F on the conjugate diameter. By shifting the paper while keeping these points fixed, you can obtain multiple points along the ellipse.
Q: What are some methods for constructing an ellipse mentioned in this chapter?
The chapter describes several methods including using a thread and pins, intersecting lines, two circles, and a paper trammel. Each method has its own advantages depending on available tools and the scale of the project.