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Hello again friends,
In-spite of including many online resources for helping you learn about math solid geometry and their manipulation, I'm getting number of comments on how to solve them; especially the question "unfolding of solids". So, in this post, I'm gonna cover their basics and will try to make you understand the process in a simpler way for your easy understanding - by considering and elaborating the process for some example. I begin with basics and then move to the topic 'How to analyse/visualize unfolding by imagination' at the end of this session.
3D solid in different views
Math solid shapes are usually viewed best in isometric drawings. View which the solid is usually 30 degree inclined with the horizontal as shown in the sample picture below are usually isometrics. They are best examples of 3D because of solid being represented in 3 co-ordinate directions.
The four common projection we come across in drawing or Engg. graphics are - front view, left side view, right side view, top view. Although bottom view is there, but it has not much significance coz most of the times, it will be a single plane surface shape.
For your additional info, let me add other types of views as taken from Wiki (note: the list includes photography views too!)
- Bird's-eye view
- Cross section
- Cutaway drawing
- Exploded view drawing
- Fisheye lens
- Worm's-eye view
- Zoom lens
Now, consider a solid shape (first angle projection), and when they are unfolded, you can get the following transformations as shown below.
unfolding of the first angle geometry
Ok, above are some basics, now let me add an example that I drew personally (in color) for your better understanding, the solid has been rotated in different directions for you to realize how it gets oriented. Also, it's important that you should note the continuation or break of surfaces in every view.
Figure also shows the direction of view, which I consider as front instead of our regular way of taking arrow in left! The following set of figures shows the same solid viewed in different 3D positions. Note that any black surface that you might see in the pics are just their shadows and has nothing to do with actual solid.
Now, the front view!
right side view is shown below, notice that the yellow box on top of the rectangular box covers the view of the blue triangle and so the triangle won't appear from left side.
Below figure shows the backside view. Observe how the view is similar to front view except the swapping of positions of the blue and yellow positions. This is not always true but for this specific geometric shape.
Below figure shows the left side view (little distorted). Actually the blue triangle width and the yellow square/retangle width are of same size and so they must lie in the same line. But because of little distortion while making the picture, it turned little. Hope you understood and don't get confused.
Ok, now that we have seen some basics, we will move to solving the question "Unfolding of solid shape" by taking an example.
Unfolding solid shapes
Below image shows the example that I considered with unique alphabets for each surface - for easy identification. I also included back view that depicts clearly the visible surfaces, such that both the views covered all the surfaces except the bottom surface, which I think is not necessary to show here as you can easily assume that.
(Note: You may want to click the image for enlarged view or to download)
Now, let's say I unfolded the surface with tag 'A', which when unfolded will assume the below shape. A is unfolded from vertical position to the horizontal position by rotating towards us as shown by the arrow in the image below (just like opening the front door of a box).
Unfold B now from vertical position to the horizontal position by rotating to the right as shown in the below image. (Note: You can unfold in any order, it really doesn't matter, I took this way. But if in exam they show options in unfolded view, then you may have to put extra effort in determining which way they unfolded. If you understand the way I proceed here, then it will be easy to identify any unfolding!)
Now, I will unfold both E and D surfaces together to vertical plane as indicated by arrow in the below image.
In the above two images, note a line tagged 'p' which will actually not be a line when A and B are unfolded but an empty space, I just included there for your visualization, in the next image, I removed that to show you how in actual the solid looks when the surfaces A and B are removed.
Now, I unfold surfaces C and F as shown in the below picture. C to the right and F to the back (indicated by arrows in the image).
Now, I'm interested in bringing all the surfaces to the ground. I mean make them flat/plain by unwrapping as indicated in the below image.
Finally, to put together, and when viewed from top, the unfolded solid looks like below
The above is only one possibility, like I mentioned there are many possibilities, I took time to draw some possibilities as I showed in the below images, can you try yourself to make some more if you get free time ? I bet you can, if you understood the above steps :)
Doesn't the above looks like opening a card box? or unwrapping your gift packs or consignments ? Okay, then consider this as your gift wrap for Pongal or new year :)
(ok, if it's late delivery, you can put fine on me :P)
I hope you enjoyed this, as well as understood how to deal with similar questions. I'm expecting your imaginative skills to be better from now especially when dealing with solid shapes and manipulations.
Good luck :)