Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Determine the line of intersection between two prisms.
• Develop the lateral surfaces of prisms after intersection.
• Construct the true shape of an intersection.
• Explain the real-world applications of these concepts.

Lesson summary

Intersection and development of surfaces are fundamental concepts in engineering graphics and design. They are crucial for creating accurate and efficient designs for a wide range of products, from packaging and sheet metal work to architectural structures and piping systems. In South Africa, these skills are essential for industries like manufacturing, construction, and renewable energy, contributing to economic growth and infrastructure development. Think of the corrugated iron roofing on many houses, or the intricate pipework in a chemical plant near Sasolburg – these all rely on accurate intersection and development.

Lesson notes

Intersection of Surfaces The intersection of two surfaces is the line (or curve) formed where they meet. Determining this line of intersection is vital for various design and manufacturing processes. For example, when joining two pipes at an angle, the shape of the cut on each pipe needs to be accurately determined to ensure a proper fit.

General Principles: Finding Points of Intersection: The key to finding the line of intersection is to identify several points that lie on both surfaces. These points are then joined to form the line (or curve) of intersection.

Auxiliary Planes: Often, auxiliary planes are used to help find these points. An auxiliary plane is a plane that intersects both surfaces. The intersection of the auxiliary plane with each surface will produce lines. Where these lines intersect, you have a point on the line of intersection between the two original surfaces.

Visibility: Remember to determine the visibility of the lines of intersection. Parts of the line that are hidden behind the object are drawn as dashed lines.

Intersection of Two Prisms: Let's consider the intersection of two rectangular prisms. This is a common and relatively simple example.

Example 1: Intersection of Two Rectangular Prisms Imagine two rectangular prisms intersecting at an angle.

Orthographic Projections: Draw the orthographic projections (front view, top view, and side view) of the two prisms. Ensure the views are aligned correctly.

Identify Key Points: Look for points where edges of one prism intersect faces of the other. These are obvious points on the line of intersection.

Auxiliary Cutting Planes: If there aren't enough obvious intersection points, introduce auxiliary cutting planes. These are usually vertical planes, parallel to either the front or side view. Choose planes that cut through the prisms, creating lines on each.

Locate Intersection Points: Where the lines created by the auxiliary plane on each prism intersect, you've found a point on the line of intersection.

Join the Points: Join the points of intersection to create the line of intersection. Remember to indicate visibility using hidden detail lines where necessary.

True Shape: To determine the True Shape of the section, project perpendicular to the line of intersection. Example showing the intersection of two prisms: Consider a vertical rectangular prism intersecting a horizontal rectangular prism. In the top view, the intersection is clear where the edges of the horizontal prism intersect the vertical prism. In the front view, however, the true shape of the cut on the vertical prism is not visible. You would have to use auxiliary views to project the true shape of the section. Development of Surfaces Development of a surface is the process of unfolding a 3D object onto a 2D plane. This is crucial for creating patterns that can be cut from sheet metal or other materials and then folded or rolled into the desired shape. This is widely used in South African workshops to create anything from ventilation ducts to metal containers.

General Principles: True Lengths: Accurate development requires knowing the true lengths of all lines involved. If a line is not parallel to the projection plane, its true length needs to be determined using rotation or auxiliary views.

Accuracy is Key: Precise measurements are essential. Even small errors can accumulate and result in a poorly fitting final product.

Development of a Prism: The development of a prism is relatively straightforward.

Example 2: Development of a Rectangular Prism Determine True Lengths: Ensure all the edges of the prism are shown in their true length. In orthographic projection, edges parallel to the viewing plane are shown in true length.

Draw the Base: Start by drawing a rectangle representing one of the bases of the prism.

Develop Lateral Surfaces: Attach rectangles to the sides of the base rectangle. The height of each rectangle corresponds to the height of the prism, and the width corresponds to the width of the respective face of the prism.

Add the Second Base: Finally, add a rectangle representing the second base to the last lateral surface.

Development of a Cylinder: The development of a cylinder is similar to that of a prism, but instead of rectangles for the lateral surfaces, you have a rectangle whose length is equal to the circumference of the cylinder.

Example 3: Development of a Cylinder Calculate Circumference: Calculate the circumference of the cylinder using the formula C = πd, where d is the diameter.

Draw the Rectangle: Draw a rectangle with a height equal to the height of the cylinder and a width equal to the circumference.

Add the Circular Bases: Add circles to either end of the rectangle, representing the top and bottom bases of the cylinder.

Account for cut-out: When a cylinder has been cut (e.g. intersects another surface) the length of the side will vary and be reduced by the height of the intersection at a particular point on the circumference.