Line Intersection & Projection: A Detailed Guide

by TextBrain Team 49 views

Hey guys! Today, we're diving into a fascinating problem involving spatial geometry: drawing a line through a given point that intersects a line segment at a specific distance from a projection plane, and then completing the frontal projection of that line given its inclination angle. Buckle up, because we're about to get technical, but I promise to keep it as straightforward as possible!

Problem Statement

Let's break down the problem into manageable chunks. We are given:

  1. A point A with coordinates (65, 25, 35).
  2. A line segment MN defined by points M(55, 5, 5) and N(10, 20, 25).

Our mission is twofold:

  • Part 1: Find a line that passes through point A and intersects the line segment MN at a point B. This point B must be 35 mm away from the profile plane of projection.
  • Part 2: Once we've found line AB, we need to construct its frontal projection, assuming we know the angle of inclination of AB.

Part 1: Finding the Intersection Point B

Understanding the Geometry

First, let's visualize what's happening. We have a point A floating in 3D space, and a line segment MN somewhere else. We need to find a line extending from A that pierces MN at a specific location, dictated by its distance from the profile plane.

Since point B lies on the line segment MN, we can express its coordinates as a parametric blend between the coordinates of M and N. Let's say point B is defined by the parameter 't', where 0 ≤ t ≤ 1. Then:

Bx = Mx + t(Nx - Mx) By = My + t(Ny - My) Bz = Mz + t(Nz - Mz)

Plugging in the values for M and N:

Bx = 55 + t(10 - 55) = 55 - 45t By = 5 + t(20 - 5) = 5 + 15t Bz = 5 + t(25 - 5) = 5 + 20t

Incorporating the Profile Plane Distance

The crucial piece of information is that point B is 35 mm away from the profile plane. In our coordinate system, this means the x-coordinate of B (Bx) is equal to 35.

So, we have:

35 = 55 - 45t

Solving for t:

45t = 55 - 35 45t = 20 t = 20 / 45 = 4 / 9

Now that we have the value of t, we can find the coordinates of point B:

Bx = 35 (given) By = 5 + 15(4/9) = 5 + 60/9 = 5 + 20/3 = (15 + 20) / 3 = 35/3 ≈ 11.67 Bz = 5 + 20(4/9) = 5 + 80/9 = (45 + 80) / 9 = 125/9 ≈ 13.89

Therefore, the coordinates of point B are approximately (35, 11.67, 13.89).

Verifying the Solution

It's always a good idea to verify our solution. Does point B lie on the line segment MN? Since 0 ≤ t = 4/9 ≤ 1, it does! Also, its x-coordinate is indeed 35, as required.

Part 2: Constructing the Frontal Projection of Line AB

Understanding Projections

Before we dive in, let's clarify what a frontal projection is. A frontal projection is a 2D representation of a 3D object projected onto a vertical plane that is parallel to the viewer. In simpler terms, it's the view you'd see if you were standing directly in front of the object.

Utilizing the Angle of Inclination

We are given the angle of inclination of line AB. This angle, often denoted as α (alpha), is the angle between the line AB and the horizontal plane. Knowing this angle is crucial for reconstructing the frontal projection.

Here's the breakdown of how we will approach this:

  1. Calculate the Horizontal Projection: The horizontal projection represents the line AB as if viewed from the top. We already have the 3D coordinates of points A and B: A(65, 25, 35) and B(35, 11.67, 13.89).

  2. Determine the Length of the Horizontal Projection: The length of the horizontal projection (AB_horizontal) can be calculated using the difference in the x and y coordinates of points A and B. We only consider x and y because the horizontal plane is defined by these two axes.

    AB_horizontal = √[(Ax - Bx)² + (Ay - By)²] AB_horizontal = √[(65 - 35)² + (25 - 11.67)²] AB_horizontal = √[30² + 13.33²] AB_horizontal = √(900 + 177.69) ≈ √1077.69 ≈ 32.83

  3. Relate the Horizontal Projection to the True Length: The relationship between the length of the horizontal projection, the true length of AB, and the angle of inclination (α) is given by:

    AB_horizontal = AB_true * cos(α)

    Therefore, we can find the true length of AB (AB_true):

    AB_true = AB_horizontal / cos(α)

  4. Calculate the Difference in Z-coordinates: The difference in the z-coordinates (Δz = Az - Bz) tells us how much the line rises (or falls) in the z-direction. This is important for defining the vertical component of the frontal projection.

    Δz = 35 - 13.89 = 21.11

  5. Construct the Frontal Projection: Now, we can construct the frontal projection. Imagine projecting the line onto the yz-plane (the plane you're looking at directly when facing the "front" of the object). In this projection:

    • The y-coordinate remains the same.
    • The z-coordinate remains the same.

  6. Representing in 2D: Therefore, to draw the frontal projection, plot the points A'(Ay, Az) = (25, 35) and B'(By, Bz) = (11.67, 13.89) on a 2D plane. Connect these points to visualize the frontal projection of the line AB.

Summary of Frontal Projection Steps

  1. Calculate the horizontal projection length (AB_horizontal).
  2. Calculate the true length of AB (AB_true) using the inclination angle.
  3. Calculate the difference in z-coordinates (Δz).
  4. Plot the projected points A'(Ay, Az) and B'(By, Bz) on the yz-plane.
  5. Connect the points to visualize the frontal projection.

Important Considerations

  • Accuracy: The accuracy of your final projection depends heavily on the precision of your calculations and measurements. Use as many decimal places as possible during intermediate calculations to minimize rounding errors.
  • Software: CAD software can greatly simplify this process. Software packages allow you to define points and lines in 3D space, and then automatically generate projections from any viewpoint.
  • Understanding Coordinate Systems: A strong understanding of 3D coordinate systems is essential for working with spatial geometry problems. Always double-check your coordinate system conventions.

Example with Assumed Angle

Let's assume the angle of inclination (α) is 30 degrees. Then, cos(30°) ≈ 0.866.

  1. We already calculated AB_horizontal ≈ 32.83.
  2. AB_true = 32.83 / 0.866 ≈ 37.91
  3. Δz = 21.11 (calculated previously)
  4. Frontal projection points: A'(25, 35) and B'(11.67, 13.89)

Now, you would plot these points on a 2D plane to visualize the frontal projection. You'd see a line sloping downwards from A' to B'.

Conclusion

This problem combines several important concepts in spatial geometry, including parametric representation of lines, coordinate systems, and projections. By breaking down the problem into smaller steps and understanding the underlying principles, you can tackle even the most complex geometric challenges. Remember to always visualize the problem in 3D space and double-check your calculations to ensure accuracy.

I hope this detailed guide helps you guys understand how to solve this problem! Let me know if you have any questions.