Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

This page supports the lesson note with a companion video and a short classroom-ready summary.

For class groups and homework, share this lesson page so learners also get the summary, objectives, and full lesson context.

Performance objectives

• Identify and describe intersections between geometric solids.
• Graphically determine the line of intersection.
• Construct the development of cylinders and cones.
• Develop truncated and intersected surfaces.
• Apply principles of accuracy and neatness.

Lesson summary

This week, we delve into the crucial topic of intersection and development of surfaces. This area of Engineering Graphics and Design is essential because it forms the foundation for understanding how complex three-dimensional objects are constructed from flat materials. Think about the corrugated iron roofs common in many South African communities, or the intricate shapes of water tanks, or even the layout of ventilation ducts in a building – all are designed and manufactured using the principles of intersection and development. A solid understanding of these principles allows us to accurately represent these shapes on paper (or digitally), enabling precise fabrication and construction.

Lesson notes

2.1 Intersection of Surfaces: When two or more surfaces meet, they create a line or curve of intersection. This line or curve represents the boundary where the surfaces join. Determining this line accurately is crucial for proper design and manufacture. The intersection can be found graphically using different methods.

Methods for Determining Intersection: Line Method: This method involves selecting lines on one surface and finding where they intersect the other surface. The points of intersection are then joined to form the line of intersection. This is particularly useful when dealing with cylinders and prisms.

Cutting Plane Method: This method involves introducing cutting planes (horizontal, vertical, or inclined) that intersect both surfaces. The intersection of the cutting plane with each surface creates lines or curves. The points where these lines or curves intersect are points on the line of intersection between the original two surfaces. The more cutting planes used, the more accurate the result.

Auxiliary View Method: In complex cases, an auxiliary view (a view projected onto a plane that is not parallel to any of the principal planes) might be necessary to simplify the intersection and make it easier to determine the line of intersection. 2.2 Development of Surfaces: Development refers to unfolding or flattening a three-dimensional surface onto a two-dimensional plane. This is essential for manufacturing items from sheet metal or other flat materials. Only developable surfaces (surfaces that can be unfolded without stretching or tearing) can be accurately developed. These include surfaces with single curvature like cylinders and cones.

Types of Surfaces and Developability: Single-Curvature Surfaces: These surfaces can be generated by moving a straight line along a curved path (e.g., cylinder, cone). They are developable.

Double-Curvature Surfaces: These surfaces have curvature in two directions (e.g., sphere). They are non-developable. Approximations are used to develop these surfaces, such as gore methods for spheres.

Development Methods: Parallel Line Development (for Cylinders): Divide the circular base of the cylinder into equal segments. Draw vertical lines upwards from each division. Transfer the circumference of the base (calculated as πd, where d is the diameter) to a straight line on the development. Divide this line into the same number of segments as the base. Draw vertical lines upwards from each of these segments. The height of the cylinder is transferred to these vertical lines, creating the development.

Radial Line Development (for Cones): Determine the true length of the slant height of the cone. This becomes the radius of the sector that forms the development. Calculate the angle of the sector using the formula: Angle = (radius of base / slant height) 360°. Draw the sector with the calculated angle and radius. This represents the development of the cone. 2.3 Worked

Examples: Example 1: Intersection of a Cylinder and a Prism (Line Method) Imagine a cylindrical water tank intersecting a rectangular prism representing a pipe connection. Draw the orthographic views (front and top) of the cylinder and prism. Select several vertical lines on the cylinder's surface in the top view. Label them 1, 2, 3, etc. Project these lines down to the front view. Identify where these lines intersect the faces of the prism in the front view. Mark these intersection points. Project these intersection points back up to the top view, onto the corresponding lines 1, 2, 3, etc. Join the points of intersection in the top view to create the line of intersection. Use hidden detail where necessary. Project the points of intersection from the top view to the front view to complete the line of intersection in the front view.

Example 2: Development of a Cylinder (Parallel Line Method) Consider a cylindrical ventilation duct with a diameter of 200mm and a height of 300mm. Calculate the circumference of the cylinder: πd = π * 200mm ≈ 628mm. Divide the circular base into 12 equal segments (30° each). Draw a horizontal line representing the developed circumference (628mm). Divide this line into 12 equal segments (628mm / 12 ≈ 52.3mm each). Draw vertical lines upwards from each segment on the horizontal line. Measure and mark the height of the cylinder (300mm) on each vertical line. Connect the top points of the vertical lines to create a rectangle. This is the development of the cylinder.

Example 3: Development of a Cone (Radial Line Method) Imagine a conical roof section with a base diameter of 300mm and a slant height of 400mm.

Calculate the radius of the base: 300mm / 2 = 150mm.

Calculate the angle of the sector: (radius of base / slant height) 360° = (150mm / 400mm) 360° = 135°. Draw a circle with a radius equal to the slant height (400mm). From the center of the circle, draw two radii, forming an angle of 135°. The sector formed by these two radii and the arc of the circle is the development of the cone.