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The image on the right shows a tea light holder based on a hyperboloid of revolution - Leaving Cert DCG - Question C-2 - 2022
Question C-2
The image on the right shows a tea light holder based on a hyperboloid of revolution. A vertical opening accommodates the tea light candle. Fig. C-2 below shows the... show full transcript
Worked Solution & Example Answer:The image on the right shows a tea light holder based on a hyperboloid of revolution - Leaving Cert DCG - Question C-2 - 2022
Step 1
Draw the outline plan and elevation of the tea light holder.
Answer
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Draw the Base and Top Circles (ISO) in Plan and Elevation:
- Start by drawing two circles in plan representing the base and the top of the tea light holder, ensuring they are concentric and properly scaled.
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Location of 'Throat' Circle in Plan and Elevation:
- In both plan and elevation, mark the location of the throat circle which represents the vertical opening.
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Construction to Determine Points on One Branch of Hyperbola in Elevation:
- Identify the points along the hypotenuse of the hyperboloid shape from the throat to the base, and use these points to outline the tapering structure in elevation.
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Determine Points on Second Branch in Elevation:
- Using symmetry, replicate the branch of the hyperbola on the opposite side for the complete elevation view.
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Draw Hyperbolic Curves in Elevation:
- Connect the points identified in the previous steps with smooth curves to represent the hyperboloid structure in elevation.
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Completion of Elevation and Plan:
- Ensure all edges and curves are clearly defined for both the elevation and plan view.
Step 2
Draw the projections of the vertical opening.
Answer
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Draw Vertical Opening in Plan:
- Sketch the vertical opening as a cylindrical shape at the specified position in the plan view, ensuring it aligns with the throat circle.
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Determine and Points on Curve in Elevation:
- In the elevation view, determine the curvature of the vertical opening by measuring the necessary height, and connecting the top and bottom of the opening with a smooth curve.
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Determine Additional Points in Elevation (min including the vertex):
- Identify key coordinates along the curve, ensuring one of them is the vertex of the hyperboloid.
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Draw Curve:
- Connect the points determined above smoothly to finalize the projection of the vertical opening.
Step 3
Determine the positions of the directrix and the focal point for one branch of the double hyperbola in the elevation.
Answer
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Construction to Determine Required Directrix:
- Begin by identifying the latus rectum of the hyperbola, then locate the directrix, which is a fixed line used to define the hyperbola attributes, at appropriate distances from the center.
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Construction to Determine Required Focal Point:
- Mark the position of the focus based on the geometric definition of the hyperbola, ensuring it is consistent with the defined directrix, completing the task by drawing both elements clearly.