Projection of Solids Cut by Simply Incline Planes (Leaving Cert DCG): Revision Notes
Projection of Solids Cut by Simply Inclined Planes
What are simply inclined planes?
A simply inclined plane is a cutting plane that appears as a straight edge when viewed in elevation. This makes it relatively straightforward to work with because you can clearly see where the plane intersects the solid in the elevation view. The cut points can be easily identified and then projected to the plan view to complete the orthographic projection.
The key advantage of simply inclined planes is their visibility in elevation view - they appear as a clear straight line, making intersection points much easier to identify compared to other types of cutting planes.
Key characteristics of simply inclined plane cuts
When a solid is cut by a simply inclined plane, the cutting plane will:
- Appear as an edge view (straight line) in the elevation
- Show clear intersection points where it meets the edges of the solid
- Create a section surface that needs to be drawn in both plan and elevation views
The beauty of simply inclined planes is that because they show as an edge in elevation, you can immediately see some of the cut points, making the drawing process more manageable.
Always remember that a simply inclined plane will only appear as a straight edge in one of your orthographic views - typically the elevation. In other views, it may appear as a surface or at an angle.
Drawing technique for sectioned pyramids
When drawing a pyramid cut by a simply inclined plane, you follow this systematic approach:
Step 1: Identify obvious points Some intersection points are immediately visible where the inclined plane meets the edges of the pyramid in elevation. These points can be directly projected down to the plan view.
Step 2: Use horizontal sections for remaining points For points that aren't immediately obvious, you use horizontal sections. This means:
- Take imaginary horizontal slices through the pyramid at different levels
- See where these slices intersect both the pyramid and the cutting plane
- Project these intersection points to find the complete cut outline
The technique works because any horizontal section through a pyramid will create a shape similar to the base, just smaller. Where this horizontal section meets the cutting plane gives you additional points on the cut edge.
Drawing technique for sectioned cones
Cones follow a similar principle but with an important difference:
Horizontal sections through cones When you take a horizontal section through a cone at any level, it will always project as a circle in plan view. This circular nature makes it easier to find intersection points because:
- You know the section will be circular
- The radius depends on the height of the section
- Where the cutting plane intersects this circle gives you points on the cut edge
Finding intersection points Just like with pyramids, some points are easily found where the cutting plane meets obvious edges in elevation. The remaining points are found using horizontal sections, but these sections will always be circular for cones.
Key Difference: Pyramids create polygonal horizontal sections (similar to their base shape), while cones always create circular horizontal sections regardless of the cutting height.
Practical drawing steps
For both pyramids and cones:
- Start with the elevation view - this shows the simply inclined plane as a straight line
- Mark obvious intersection points where the plane meets visible edges
- Project these points down to the plan view
- Use horizontal sections to find additional points as needed
- Complete the section outline in both views
- Shade or hatch the cut surface to show it clearly
Always work systematically from the most obvious points to the less obvious ones. This prevents errors and ensures you don't miss any critical intersection points.
Understanding the views
The orthographic projection system uses multiple views to fully describe the 3D object:
- Plan view: Looking down from above, shows the horizontal cross-section
- Elevation view: Looking from the side, shows the height and the cutting plane as an edge
- Section views: Show what the inside of the cut solid looks like
Each view provides essential information that helps you understand the complete 3D form of the sectioned solid.
Key Points to Remember:
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Simply inclined planes appear as straight edges in elevation, making intersection points easier to identify
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Use horizontal sections to find intersection points that aren't immediately obvious in the elevation view
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Pyramids create polygonal sections when cut, while cones create circular or elliptical sections
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Always project points between plan and elevation views to ensure accuracy
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Shade the cut surfaces to clearly show where the solid has been sectioned by the inclined plane