Lesson Notes By Weeks and Term v3 - Senior Secondary 2
Auxiliary views of Geometrical Solids
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Auxiliary views of Geometrical Solids
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Subject: Technical Drawings
Class: Senior Secondary 2
Term: 1st Term
Week: 5
Theme: Pictoral Drawing
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An auxiliary view is an orthographic projection of an object onto a plane that is not one of the principal projection planes (horizontal, frontal, or profile). This special projection plane, called an auxiliary plane, is positioned parallel to a specific inclined surface of the object. The primary purpose of an auxiliary view is to show the true shape and size of an inclined surface, which appears foreshortened or distorted in the standard orthographic views. Why are auxiliary views necessary? When an object has a surface that is inclined to one or more of the principal planes, its projection onto these principal planes will not show its true dimensions.
The construction of an auxiliary view follows specific principles: Identification of the Inclined Surface: The first step is to identify the inclined surface whose true shape and size are required.
Orthographic Views: The object must first be drawn in its standard orthographic views (e.g., front elevation, plan, and/or end elevation). These views provide the necessary information for projection.
Reference Line (H/F or X1Y1 Line): A reference line, often labeled X1Y1 (or H/F if projecting from a horizontal plane, or F/P if projecting from a frontal plane), is drawn parallel to the edge view of the inclined surface in one of the principal views. If the inclined surface appears as a line in the front elevation, the X1Y1 line is drawn parallel to this line to project an auxiliary elevation. If the inclined surface appears as a line in the plan view, the X1Y1 line is drawn parallel to this line to project an auxiliary plan.
Projection Lines: Projectors are drawn perpendicular to the X1Y1 reference line from every key point on the object in the principal view from which the auxiliary view is being projected. These projectors extend towards the auxiliary plane.
Transfer of Dimensions: Distances perpendicular to the X1Y1 line are transferred from the adjacent principal view. When projecting an auxiliary elevation from the front elevation (meaning the X1Y1 line is parallel to the inclined line in the front elevation), the depths (distances from the X-Y line in the plan view) are transferred. When projecting an auxiliary plan from the plan view (meaning the X1Y1 line is parallel to the inclined line in the plan view), the heights (distances from the X-Y line in the front elevation) are transferred. These transferred distances are measured from the main reference line (X-Y) of the principal views to the corresponding points.
Connecting Points: Once all key points are located on the auxiliary plane, they are connected in the correct order to form the auxiliary view, showing the true shape and size of the inclined surface. Visible lines are drawn bold, and hidden lines are dashed. For SS2, the focus is primarily on Primary Auxiliary Views: Auxiliary Elevation: Projected from the plan view, usually to show true heights or when an inclined surface appears as a line in the plan.
Auxiliary Plan: Projected from the front elevation or end elevation, to show true depths or when an inclined surface appears as a line in the elevation.
Active Listening and Participation: Listen attentively to the teacher's explanations and demonstrations. Ask questions for clarification.
Instrument Preparation: Ensure all necessary drawing instruments (drawing board, T-square, set squares, pencils (HB, 2H), eraser, compass, divider, ruler, drawing sheets) are ready.
Observation and Note-taking: Observe teacher demonstrations carefully, noting key steps and principles in their exercise books.
Practical Drawing: Engage in guided practice activities, drawing orthographic and auxiliary views as demonstrated by the teacher.
Problem Solving: Work independently or in small groups on assigned practice problems, applying the learned techniques.
Self-Assessment: Compare their drawings with the teacher's examples or provided solutions, identifying areas for improvement. is inclined at 60 degrees to the H
P. Draw its: a) Front Elevation b) Plan c) Auxiliary View showing the true shape of the cut surface.
Solution 3: (a)
Front Elevation:
1. Draw X-Y line.
2. Draw an isosceles triangle with base 60mm on the X-Y line and height 80mm. This is the cone's front elevation.
3. Divide the base into 8-12 equal parts for accuracy (and project these divisions up to the apex to generate generators, if needed for complex curves, but for this basic cut, not strictly necessary for the elevation).
4. Mark a point on the axis 40mm from the base.
5. Draw a line through this point, inclined at 60 degrees to the horizontal (X-Y line), cutting the two slant sides of the cone. Label the intersection points P and Q on the slant sides.
6. Darken the visible lines of the truncated cone below the cutting plane. (b)
Plan:
1. Project downwards from the front elevation.
2. Draw a circle of 60mm diameter for the base.
3. Project points P and Q from the front elevation to the plan. Note that P and Q are on the slant generators. To accurately project them, draw horizontal projection lines from P and Q in the front elevation to the cone's outline, then project vertically down to the corresponding points on the circle (base) in the plan. This will result in an elliptical shape for the cut surface in the plan. (c) Auxiliary View (True Shape of Cut Surface):
1. Identify the edge view of the cutting plane: In the front elevation, the line P-Q represents the cutting plane (it appears as a line because it's perpendicular to the VP).
2. Draw the Auxiliary Reference Line (X1Y1): Draw a line X1Y1 parallel to the cutting plane line P-Q in the front elevation.
3. Draw Projection Lines: From point P, Q, and any other points used to define the curve of the cut (e.g., intermediate points on the cutting plane in the front elevation), draw projection lines perpendicular to the X1Y1 line.
4. Transfer Depths from Plan: From the plan view, measure the perpendicular distances from the X-Y line (of the principal views) to each of the projected points (P, Q, and intermediate points). For example, if you divide the base into 12 parts, project these to the front elevation. The cutting plane will cut through these generators. Project these points down to the plan to find their depths. Transfer these depths along the projection lines, measuring from the X1Y1 line.
5. Connect the Points: Connect the transferred points with a smooth curve to obtain the true elliptical shape of the cut surface.
Commentary: Obtaining the true shape of an elliptical cut on a cone requires careful projection of multiple points (usually 8-12) along the cutting plane to ensure accuracy of the curve. This is an excellent example of where auxiliary views are indispensable.
Problem: Draw the front elevation, plan, and an auxiliary elevation showing the true shape of the inclined surface of the wedge-shaped block shown below. (Assume standard third-angle projection). (Imagine a drawing of a wedge: a rectangular prism with one top corner cut off at an angle, forming an inclined rectangular face. Let the front face be vertical, the base horizontal, and the inclined surface slanting from top-left to bottom-right in the front view.)
Dimensions: Length (left to right): 60mm Width (front to back): 40mm Height (left edge): 50mm Height (right edge): 20mm The inclined surface connects the top of the 50mm height to the top of the 20mm height.
Step-by-step Solution: Draw the Front Elevation: Draw a horizontal line (X-Y line). Above X-Y, draw a rectangle 60mm long and 50mm high (representing the left vertical face). From the top-right corner of this rectangle, measure down 30mm (50-20=30) and mark a point. This is the top-right point of the 20mm height. Connect the top-left corner (height 50mm) to the marked point (height 20mm). This inclined line represents the inclined surface. Complete the rest of the front elevation (base and vertical right edge). Label points (e.g., A, B, C, D for base, A', B', C', D' for top).
Draw the Plan: Project vertical lines downwards from the front elevation to the plan view. Draw another horizontal line (X-Y line for plan) below the elevation. Draw a rectangle (60mm long, 40mm wide) representing the base of the block. The inclined surface will appear as a rectangle or a line depending on the projection. In the plan, the inclined surface will appear as a rectangle of 60mm length and 40mm width. It is not true shape in this view. Project the Auxiliary Elevation (True shape of inclined surface): Identify the inclined edge: In the front elevation, the inclined surface appears as a single inclined line (let's say A'D').
Draw the Auxiliary Reference Line (X1Y1): Draw a line (X1Y1) parallel to the inclined line A'D' in the front elevation. Place it at a convenient distance away from the front elevation.
Draw Projection Lines: From each corner point of the inclined surface in the front elevation (A', D', and the hidden points below them), draw projection lines perpendicular to the X1Y1 line.
Transfer Depths from Plan: Establish a new datum on the X1Y1 line. From the main X-Y line of the front elevation, choose a reference point (e.g., the front edge of the block). In the plan view, measure the perpendicular distance from the main X-Y line to each corresponding point on the object. These are the depths. Transfer these depths along the projection lines, measuring from the X1Y1 line. For example, if a point is 10mm from the X-Y line in the plan, mark it 10mm from the X1Y1 line along its respective projector. For the example wedge block, assume points 1, 2, 3, 4 form the inclined surface. Points 1 and 2 are on the front edge (depth = 0 from the front), points 3 and 4 are on the back edge (depth = 40mm from the front). Measure 0mm from X1Y1 for points 1, 2 and 40mm from X1Y1 for points 3,
4. Connect the Points: Connect the transferred points to form the true shape of the inclined surface. This will be a rectangle, representing the true shape of the inclined face of the wedge.
Architectural Design and Construction (Roofing & Ramps): In Nigeria, auxiliary views are critical for designing complex roof structures, such as hip roofs or gambrel roofs, prevalent in residential and commercial buildings. Standard orthographic views cannot show the true shape and size of each inclined roof panel. An auxiliary view provides the true dimensions, which are essential for calculating material quantities (e.g., roof sheets, timber rafters) and for accurate cutting and assembly by artisans. Similarly, designing access ramps for buildings, especially for people with disabilities, requires accurately determining the true slope and surface area, which auxiliary views facilitate. Mechanical Engineering and Fabrication (Machine Parts): Many machine components, particularly those produced in local fabrication workshops or industrial settings, feature inclined surfaces (e.g., chamfers, tapered shafts, angled brackets, or components of agricultural machinery). For example, the design of a specialized blade for a local agricultural machine or an angled mounting bracket requires the true dimensions of its inclined faces to ensure correct fit, strength, and manufacturing precision. Auxiliary views allow engineers and draughtsmen to create accurate manufacturing drawings that guide machining and assembly processes. Furniture Design and Craftsmanship (Angled Furniture): Modern Nigerian furniture designs often incorporate aesthetic angles and inclined surfaces (e.g., tilted chair backs, angled table legs, sloping cabinet fronts). For skilled carpenters and furniture designers, auxiliary views are invaluable for developing precise cutting lists and templates for these angled components. This ensures that pieces fit together perfectly and the finished product has the intended aesthetic and structural integrity, improving the quality of locally made furniture.