Lesson Notes By Weeks and Term v3 - Senior Secondary 2
Planes and Views in Space
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Planes and Views in Space
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Subject: Technical Drawings
Class: Senior Secondary 2
Term: 1st Term
Week: 10
Theme: Points And Line In Space
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Explain the terms in the planes and views in space Place line in the first quadrant and project in to the vertical and horizontal planes.
Points And Line In Space Planes and Views in Space Term: 1st Term Week: 10 ---
1. Overview and Learning Objectives This topic introduces students to the fundamental principles of orthographic projection, specifically focusing on the representation of lines in space using the First Angle Projection method. It lays the groundwork for understanding how three-dimensional objects are accurately depicted in two dimensions, a critical skill for various technical professions. The concepts covered are essential for interpreting and creating technical drawings in fields such as engineering, architecture, and manufacturing, which are vital for Nigeria's infrastructural development and industrial growth.
Learning Objectives: By the end of this lesson, students will be able to:
1. Define and explain the key terms associated with planes and views in space, such as Horizontal Plane (HP), Vertical Plane (VP), Ground Line (GL), and First Angle Projection.
2. Accurately position a line segment within the first quadrant of projection.
3. Correctly project the front view (elevation) and top view (plan) of a line onto the Vertical Plane (VP) and Horizontal Plane (HP), respectively.
4. Determine the true length and true angles of inclination (with HP and VP) of a line from its projected views. Connection to Real-World Applications in Nigeria: Understanding planes and views is foundational for: Architecture and Construction: Interpreting building plans, sections, and elevations for residential houses, commercial buildings, and infrastructure projects like bridges and flyovers common in Nigerian cities.
Mechanical and Civil Engineering: Designing and manufacturing machine components, vehicle parts (e.g., in Nnewi's automotive cluster), and civil structures.
Surveying and Cartography: Creating accurate maps and land survey plans, essential for urban planning and resource management in Nigeria.
Product Design: Developing prototypes for products ranging from furniture to consumer electronics, ensuring functionality and aesthetic appeal.
2. Key Concepts and Explanations A. Orthographic Projection Orthographic projection is a method of representing three-dimensional objects in two dimensions by projecting views of the object onto two or more projection planes, using parallel projectors perpendicular to the planes. In Nigeria, the First Angle Projection method is predominantly used, aligning with British and European standards.
B. Planes of Projection For representing objects in space, two principal planes are generally used: Horizontal Plane (HP): An imaginary plane representing the horizontal surface. When looking down on an object, its view is projected onto the H
P. This view is known as the Top View or Plan.
Vertical Plane (VP): An imaginary plane representing a vertical surface. When looking at an object from the front, its view is projected onto the V
P. This view is known as the Front View or Elevation.
Ground Line (GL) or X-Y Line: This is the imaginary line where the Horizontal Plane (HP) and Vertical Plane (VP) intersect. It serves as a reference line on the drawing sheet, separating the area for the front view and top view after the HP is rotated. C. Quadrants of Projection The HP and VP divide space into four quadrants. In First Angle Projection, the object is always placed in the First Quadrant.
First Quadrant: The space above the HP and in front of the VP. When viewed from the front, the object's front view (elevation) is projected onto the VP. When viewed from the top, the object's top view (plan) is projected onto the HP. For drawing on a 2D sheet, the HP is imagined to rotate downwards by 90° about the GL until it lies in the same plane as the VP. This places the top view below the GL and the front view above the G
L. D. Views in First Angle Projection (for a Line)
Front View (Elevation): The projection of the line onto the VP. It shows the length and height dimensions of the line. The length of the front view (elevation) is denoted as a'b'.
Top View (Plan): The projection of the line onto the HP. It shows the length and width (or depth) dimensions of the line. The length of the top view (plan) is denoted as ab.
Projectors: Imaginary lines that extend from every point of the object perpendicular to the projection plane. For a line AB, projectors A-a, Line)
Front View (Elevation): The projection of the line onto the VP. It shows the length and height dimensions of the line. The length of the front view (elevation) is denoted as a'b'.
Top View (Plan): The projection of the line onto the HP. It shows the length and width (or depth) dimensions of the line. The length of the top view (plan) is denoted as ab.
Projectors: Imaginary lines that extend from every point of the object perpendicular to the projection plane. For a line AB, projectors A-a, B-b are used for the top view, and A-a', B-b' for the front view. On the drawing sheet, projectors linking 'a' to 'a'' and 'b' to 'b'' are perpendicular to the GL. E. Placing a Line in the First Quadrant and its Projections To place a line AB in the first quadrant, the positions of its endpoints A and B are defined relative to HP and V
P. Example: A point P is 30mm above HP and 20mm in front of VP. Its front view p' will be 30mm above the GL on the VP. Its top view p will be 20mm below the GL on the HP (after rotation). * A vertical line (projector) connects p and p', passing through the GL. F. Determining True Length and True Angles of a Line When a line is inclined to both HP and VP, its projected lengths (front view and top view) will be shorter than its actual or true length. Similarly, the angles its projections make with the GL are not its true angles of inclination. To find the true length (TL) and true angles (θ with HP, Φ with VP), the Revolution (Rotation) Method is commonly used in SS
2. Revolution Method for True Length (TL) and True Angles (θ, Φ): Case 1: True Length and True Angle with HP (θ)
1. Draw the projections: First, accurately draw the front view (a'b') and top view (ab) of the line, ensuring the projectors a-a' and b-b' are perpendicular to the GL.
2. Make the top view parallel to GL: From one end of the top view (e.g., 'a'), draw an arc with radius equal to the top view length (ab) such that the new position (e.g., 'b1') lies on a horizontal line through 'a' and is parallel to the GL. This new line (ab1) represents the line in a position where it is parallel to the VP, and its front view will show the true length.
3. Project to Front View: From 'b1', draw a projector perpendicular to the GL.
4. Locate True Length: From 'b'' (the corresponding end in the front view), draw a horizontal line parallel to the GL, intersecting the projector from 'b1' at a point 'b1''.
5. Draw True Length: Join 'a'' to 'b1''. The length a'b1' is the True Length (TL) of the line.
6. True Angle with HP (θ): The angle that this True Length line (a'b1') makes with the horizontal line drawn from 'a'' is the True Angle (θ) of the line with the H
P. Case 2: True Length and True Angle with VP (Φ)
1. Draw the projections: (Same as step 1 above).
2. Make the front view parallel to GL: From one end of the front view (e.g., 'a''), draw an arc with radius equal to the front view length (a'b') such that the new position (e.g., 'b2'') lies on a vertical line through 'a'' and is parallel to the GL. This new line (a'b2') represents the line in a position where it is parallel to the HP, and its top view will show the true length.
3. Project to Top View: From 'b2''', draw a projector perpendicular to the GL.
4. Locate True Length: From 'b' (the corresponding end in the top view), draw a vertical line parallel to the GL, intersecting the projector from 'b2'' at a point 'b2'.
5. Draw True Length: Join 'a' to 'b2'. The length ab2 is the True Length (TL) of the line. (
Note: TL from Case 1 and Case 2 must be the same).
6. True Angle with VP (Φ):** The angle and its top view will show the true length.
3. Project to Top View: From 'b2''', draw a projector perpendicular to the GL.
4. Locate True Length: From 'b' (the corresponding end in the top view), draw a vertical line parallel to the GL, intersecting the projector from 'b2'' at a point 'b2'.
5. Draw True Length: Join 'a' to 'b2'. The length ab2 is the True Length (TL) of the line. (
Note: TL from Case 1 and Case 2 must be the same).
6. True Angle with VP (Φ): The angle that this True Length line (ab2) makes with the vertical line drawn from 'a' is the True Angle (Φ) of the line with the V
P. Summary of True Length Determination (using one method and applying principles): Essentially, to determine true length, one projection of the line (e.g., top view) is rotated until it is parallel to the GL (meaning the line is parallel to the other plane, VP). The corresponding new projection (e.g., front view) will then show the true length. The angle this true length makes with the reference line (horizontal for HP, vertical for VP) in that rotated projection is the true angle.
Example Scenario: Imagine an inclined roof truss member in a building under construction in Abuja. Its actual length and inclination angles are crucial for structural integrity, but its representation on a 2D plan might not show these directly. Technical drawings allow engineers to derive these true dimensions.
3. Teaching and Learning Activities
A. Teacher Activities:
1. Introduction and Recap (5 min): Begin by briefly reviewing the concept of points in space and their projections from the previous lesson. Explain that this lesson extends to lines.
2. Concept Explanation (15 min): Introduce the terms HP, VP, GL, and First Angle Projection using clear diagrams on the whiteboard or through a projector. Demonstrate the conceptual rotation of the HP around the GL to bring it into the plane of the VP, illustrating how the top view appears below the GL and the front view above. Explain the quadrants, emphasizing the First Quadrant for First Angle Projection. Use a physical model (e.g., two hinged cardboard sheets for HP and VP, and a pencil or ruler as a line) to visually demonstrate projection.
3. Demonstration of Line Placement and Projections (15 min): Provide a specific example of a line with given endpoint coordinates (e.g., A is 10mm above HP, 20mm in front of VP; B is 40mm above HP, 50mm in front of VP). Demonstrate step-by-step on the whiteboard how to draw the front view (a'b') and top view (ab) using projectors perpendicular to the G
L. Emphasize neatness and accuracy.
4. Demonstration of True Length and True Angles (20 min): Using the projected views from the previous demonstration, guide students through the Revolution Method to determine the True Length (TL) and True Angles (θ and Φ). Carefully illustrate the steps of rotating the top view parallel to GL to find TL and θ. Similarly, illustrate rotating the front view parallel to GL to find TL and Φ. Stress that the two calculated True Lengths must be identical.
5. Guided Practice (10 min): Lead students through a simplified example, providing hints and correcting errors as they attempt to sketch on their drawing pads.
6. Q&A and Clarification (5 min): Address any student questions or misconceptions.
B. Student Activities:
1. Active Listening and Note-taking: Students will pay attention to explanations and take concise notes on definitions and procedural steps.
2. Observation: Observe the teacher's demonstrations on the whiteboard/projector and with physical models.
3. Sketching/Drawing: Students will practice drawing the projections and determining true lengths/angles as demonstrated by the teacher, using their drawing instruments.
4. Participation: Engage in discussions, ask questions for clarification, and provide answers when prompted.
5. Measurement: Accurately measure distances and angles as required for drawing and determining true dimensions.
Materials: Whiteboard/chalkboard and markers/chalk Technical Drawing instruments (T-square/set squares, protractor, compass, pencils)
Drawing sheets Physical model: two hinged cardboard/plywood sheets representing HP and VP, and a pencil/ruler to represent a line in space. * Projector