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A design for a patio fireplace based on a regular solid is shown - Leaving Cert DCG - Question A-3 - 2017
Question A-3
A design for a patio fireplace based on a regular solid is shown. The drawing shows the elevation and plan of a similar solid. All faces are regular pentagons. (a... show full transcript
Worked Solution & Example Answer:A design for a patio fireplace based on a regular solid is shown - Leaving Cert DCG - Question A-3 - 2017
Step 1
Using the given X,Y,Z, draw the true length of the line of intersection between faces A and B, and hence determine the dihedral angle between the two faces.
Answer
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Construct True Length: Start by using the horizontal reference planes X and Y. Find the true length of the line of intersection by projecting the edges of faces A and B onto the vertical plane.
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Identify Intersecting Points: Locate the points where the edges from face A intersect with those from face B.
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Draw True Length: Draw a line connecting these intersecting points on the reference planes, ensuring it reflects the actual scale and orientation between faces A and B.
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Calculate Dihedral Angle: To find the dihedral angle, use trigonometric relations or geometric constructions to measure the angle between the planes at the line of intersection. Typically, the dihedral angle can be determined using the formula: heta = an^{-1} rac{h}{d} where h is the height and d is the horizontal distance between the projections.
Step 2
Determine the centroid of face B in plan and draw the projections of a sphere which rests on the horizontal plane and is also in tangential contact with the centre point of face B.
Answer
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Find Centroid of Face B: Identify the vertices of face B from the drawing. The centroid can be calculated by averaging the vertices’ coordinates: C_x = rac{x_1 + x_2 + ... + x_n}{n}, C_y = rac{y_1 + y_2 + ... + y_n}{n} where n is the number of vertices.
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Projecting the Sphere: Draw a horizontal plane at the base of the fireplace design. Place the calculated centroid on this plane.
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Determine Sphere Radius: Select an appropriate radius such that the sphere touches the horizontal plane and the centroid of face B.
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Draw Projections: Using the center and radius, draw the projections of the sphere from the centroid. Ensure the sphere’s dimensions reflect accurate perspective with respect to the horizontal plane.