Sketchbook: Perspective Puzzles

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    Perspective Puzzles

    Welcome! The overall idea behind this thread is very simple: we all study perspective and the problems that lie with it (constructions/orthographic projections/renderings/and much more). As it develops, I'll be posting up assignments and "puzzles" for people to try out that are suggested to me so as to make this stuff fun as well as educational. However as of yet I have nothing, so I'd love some ideas as to what everyone would like to study in here, so please, post your comments letting me know what you'd want everyone to learn!

    (I'll just post a simple construction/rendering exercise to give an example of what will possibly be studied in here).

    Edit: I've gotten questions from people asking if they can join this thread, so I'll clarify here: this thread is for anyone and everyone who'd like to participate. We all need to learn this stuff, so we're all here to help each other. So please, whether you've just started learning perspective or you've been a master of it for years, feel free to contribute what you'd like. I only ask that this thread stay on topic so as to keep a productive environment. Thanks.

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    Last edited by MrFrenik; February 25th, 2013 at 11:42 PM.
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    this is gonna be awesome,to start it off I thought of a little construction challenge which isnt that hard.
    its basically an s curve on a plane mirrored downwards and mirrored perendicularly on either side.
    here is an example

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    try drawing it freehand without too many aids


    *I think you should change the order of the picture in the first post because that firs picture thumbnail is invisible.

    Last edited by GrayPersona; February 20th, 2013 at 01:27 PM.
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    Looking forward to seeing this series progress! may i recommend some form studies where you combine several different random forms and shapes and show how they merge into one another? like show how two cones would collide and join into 1 shape.

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    Awesome man!!! Sorry for my inactivity latley, about to start posting again Ill do some stuff for this as well. I think we should make a facebook page as well so we have an album with the tutorials in it. Just random thoughts hahah

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    Sooooo im no master hahahahaha but hopefully this helps a bit Here is my first ever attempt at a tutorial hahahahah The Star CLass Battleship-1000!!!!!! mwhahahahaha
    Also sorry for miss spelling some words like Vertical hahahahah


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    16 is suppose to say "shadow" not "shaw" hahahahaha

    Last edited by Joseph Church; February 21st, 2013 at 03:30 AM.
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    This is awesome, guys. I'm already happy about the participation we're getting. So far we have curved planes constructions, boolean operations (merging forms into one another, as suggested by turtle), and a kick ass tutorial by Joe on form construction. These are great and I'll definitely be thinking of my own over the next few days, since I'll be out of town until Sunday night. I'll post some stuff up then, but be sure to keep going with ideas!

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    I've taken Joe's design, redrew it from a different perspective (wide angled lens at mid horizon) and then did a cast shadow construction. Posted here are images of my process (I'll post a more in depth description of what I did when I have time, so don't fret). I've also posted a Sketchup model, which I used to "check" my cast shadows' correctness and plausibility. Let me know of any questions you might have.

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    dude thats great,probably will try to do this tomorrow.

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    this thread is awesome i shall participate. though, im not to amazing with straight lines :s

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    Thanks for the thread. I'm just now beginning to study perspective via Norling. And this will be great to visit and work with. I already know Joe777k7 has a good grasp on it, and MrFrenik post 7 , wow. Nicely done! I have yet to determine through Norlings book, or even reading threads here on CA.org how to manage off the page VP. Vertical horizon, but nothing short of just guessing. Anyone point me in a direction for this. Hopefully somewhere that breaks it down for illiterate dyslexic math deficient clowns like myself, because it's truly hard for me to grasp. I do have handprint to go through still. Anyways, If you have some simple exercises I'd love to have at them. Even though they appear to be no-brainers for you lot. Look forward to more here regardless.

    Last edited by I STRaY I; February 26th, 2013 at 01:55 PM.
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    tried my hardest to do this today and I still think there are some mistakes,if anyone sees them please tell me,I especialy found it hard to construct the cast shadow from the turret to the body of the ship.

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    Quote Originally Posted by I STRaY I View Post
    Thanks for the thread. I'm just now beginning to study perspective via Norling. And this will be great to visit and work with. I already know Joe777k7 has a good grasp on it, and MrFrenik post 7 , wow. Nicely done! I have yet to determine through Norlings book, or even reading threads here on CA.org how to manage off the page VP. Vertical horizon, but nothing short of just guessing. Anyone point me in a direction for this. Hopefully somewhere that breaks it down for illiterate dyslexic math deficient clowns like myself, because it's truly hard for me to grasp. I do have handprint to go through still. Anyways, If you have some simple exercises I'd love to have at them. Even though they appear to be no-brainers for you lot. Look forward to more here regardless.

    Stray, I drew up some steps to help you. Funny enough this type of "estimating" uses quite a bit of construction, but it's still a good way to check your estimates if you're unsure. This method also only works in 2 point perspective, since non-converging vertical lines are used as guides for plotting points. Let me know if any of it's too confusing or if I need to explain something more in depth or better.


    0. Establish the Horizon Line as well as your converging parallel lines, shown here as red and blue lines. After this you will establish 3 more parallel, perpendicular lines to your horizon: 1 at the left, 1 and the right, and once at the midpoint through which your lines cross. These lines can really be any distance on the page, as long as you have a left one that will act as your left vp, a right one that will act as the right, and a midline which both your vp's will share. In this example, my blue line is headed towards the right vp somewhere off the page, and my red line is towards the left vp. Each line passes through 2 points, 1 through its respective vp guide line and one through the shared midline. You'll notice that the converging colored lines do touch the other vp's guide line across from it, but this has nothing to do with the construction and only occurs because I've carried the lines out across the page.

    1. The next step is to establish some 1/2 way points on each of your lines. These points will half the distance between your already established points and points at which these guide lines cross the Horizon Line.

    2. Now it's a simple matter of connecting the dots. I've drawn another red line going through the 1/2 way points of both the left guide line and the midline, and I've also drawn a blue line going through the right guide line and the midline. At this point it's starting to look like a perspective grid.

    3. Now that we have a distance between dots on the lines, we can continue to take this out as far as we'd like. I've added a few more for each line. It's critical to remember to add the SAME AMOUNT of points to each line, since certain points have to go with certain points.

    4. More connecting lines, and now we have a decent, estimated grid to use for whatever we may need.

    Hope this helps!

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    Last edited by MrFrenik; February 26th, 2013 at 04:48 PM.
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    Gray, the construction looks great, man. I haven't checked it for complete accuracy, but from glancing at your guidelines and the end result it looks pretty much correct to me. Casting shadows from elevated planes can be tricky, but all you need to do is think of is the ship as a "ground plane" that your turret sits on, and then it's no different in construction method from anything else you did. If the shadow extends past this plane and onto a lower plane, then you'll have to extend your shadow points down to that plane, draw guide lines from the SVP out through those lines and then find where guide lines from your LVP intersect those. A bit more construction, but not too difficult. In hindsight, it probably would've been a better example had I done an angle that showed off the boat from the top so you could see how the cast shadow fell across the boat, but your example worked well. Keep it up!

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    Oh man, so rad, thanks MrFrenik. I think your explanation was nails. I gave it a go, and afterwards experimented with an even smaller scale on the same horizon using the same method, knowing one sets VP would/should land visible on the Horizon line, and after finished I checked to see if it would and they were pretty close to the same vanishing point. I'm so surprised I didn't get this prior, but having seen it and explained along side is so much easier,for me at least. Anyways here's my attempt, I think I did okay, and I look forward to more practice, less confusion. Name:  perspeccise2_26_13.jpg
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    Stray, it turned out great, and I'm glad that it was easy to follow for you. Keep posting your work, guys! Let's keep this thing alive.

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    ( Click to show/hide )
    Quote Originally Posted by MrFrenik View Post
    Stray, I drew up some steps to help you. Funny enough this type of "estimating" uses quite a bit of construction, but it's still a good way to check your estimates if you're unsure. This method also only works in 2 point perspective, since non-converging vertical lines are used as guides for plotting points. Let me know if any of it's too confusing or if I need to explain something more in depth or better.


    0. Establish the Horizon Line as well as your converging parallel lines, shown here as red and blue lines. After this you will establish 3 more parallel, perpendicular lines to your horizon: 1 at the left, 1 and the right, and once at the midpoint through which your lines cross. These lines can really be any distance on the page, as long as you have a left one that will act as your left vp, a right one that will act as the right, and a midline which both your vp's will share. In this example, my blue line is headed towards the right vp somewhere off the page, and my red line is towards the left vp. Each line passes through 2 points, 1 through its respective vp guide line and one through the shared midline. You'll notice that the converging colored lines do touch the other vp's guide line across from it, but this has nothing to do with the construction and only occurs because I've carried the lines out across the page.

    1. The next step is to establish some 1/2 way points on each of your lines. These points will half the distance between your already established points and points at which these guide lines cross the Horizon Line.

    2. Now it's a simple matter of connecting the dots. I've drawn another red line going through the 1/2 way points of both the left guide line and the midline, and I've also drawn a blue line going through the right guide line and the midline. At this point it's starting to look like a perspective grid.

    3. Now that we have a distance between dots on the lines, we can continue to take this out as far as we'd like. I've added a few more for each line. It's critical to remember to add the SAME AMOUNT of points to each line, since certain points have to go with certain points.

    4. More connecting lines, and now we have a decent, estimated grid to use for whatever we may need.

    Hope this helps!


    Man that was pretty elegant. Didn't think of doing that. I'll be sure to use it sometime.

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    Orthographic Projection Exercise

    So here's a simple step-by-step of setting up a 2pp view based on a simple orthographic plan. This method uses plans and elevations, which I'll explain as well as I can.

    1. I've used a squared grid to draw up a simple shape. They arranged in a specific manner, so this is important to notice. The top view is to the top left, the front is directly below the top and the right is to the right of the front. You'll notice that the width of the top view and front views are exactly the same, the height of the front and side views are the same and the length of the top and side views are the same. This is crucial to make sure of, as anything else will result in an object that doesn't translate into perspective. The first step into transferring our object is to set up a Horizon Line and placing the top view of our object with one of its corners touching our HL, essentially rotating the object. The rotation is entirely arbitrary, but for the best presentation of your object it's important to rotate it so that the most information/detail about your object will be shown. After this, I've run two sets of parallel lines from my object, shown in red and blue. These are the lines that will be used in the next step to establish my vanishing points.

    2. I now establish my Station Point, or the point at which the viewer is standing in relation to the object being viewed. You can imagine looking down on this scene, the black dot being a person staring at the corner closest to him of some distant object. From this dot, we run lines that parallel our blue and red lines until they intersect with the HL. This will give us our left and right Vanishing Points. If you'd like to check the accuracy of your lines, you can pull out a protractor and measure the angle made between LVP/RVP. It will ALWAYS be 90 degrees.

    3. Moving along, I've established a ground line, or the place at which my front corner of my object will be touching. This line is arbitrary. After this I've set up my "elevations", which are just my front and side views. These are placed on either side of my SP (station point) line and are matched to be at exactly the same height. I'll say it again: they HAVE to be the same height as one another, or else certain things will not transfer correctly.

    3a. Now onto our measured lines from our plan view, or top view. I want to transfer the square grid from my top, front and side views, so in order to do that I have to run lines from each of the points of my top view down to my station point. Where it intersects the HL is where I'll drop verticals down to give me all the verticals in my object. This is shown in the next step.

    4. Having found my vertical intersections, I now drop straight lines down from those points to my ground line. You'll also notice that I've drawn converging lines from my front corner. These are my main guide lines. From there I find where my vertical lines intersect my guide lines and continue to draw converging lines from these points. This gives me my square grid. This can probably be sort of confusing, so let me know if it needs clarifying.

    5. I now continue to develop my overall box shape. This starts with two more converging lines from the top corner (I have my height because of the front and side views) and then bring up lines from my ground plane to complete the box.

    6. Now I want to add divisions to my box, so I've drawn light guides from my front and side views through my front corner line ( which is also called the "true height line", since this is the only line in 2pp from which you can do accurate measuring) and then I extend lines to both VP's. This divides the box nicely for me.

    7. It gets messy here, and I'm sorry for that, but because my front corner is actually 1 tile "inside" my object, there is a whole row that I had to account for past my ground line. This involves finding midpoints of my closest ground square and then duplicating it forward so that I can extend my box 1x5x4 forward. I hope this doesn't throw anyone off too much. Let me know and I'll explain it better if it does. But after all that, I roughed out my object using the plans/elevations as guides. This can be tricky, as you have to actively think about where certain squares exist and where certain ones are missing in your object. It's all part of the puzzle, which can be fun but can also get frustrating.

    8. Clean lines and lineweight make for a decent presetnation!

    Hope you enjoy!

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    Ortho Assignment

    And for those of you wanting to try, here are a few examples taken from an engineering course's website. I'll place this here to give the site credit, but don't peek unless you want the answers!
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    Awesome MrFrenik, thanks for the new stuff. I've finished the boat, tried to go for a worms eye but almost came out the same as yours. Also, not sure I got the nose right, or the shadows, or a few other things but it's close enough I think. Also did Grey's S curve one. Not so much freehand except for the curves, just want to get measuring and placement/plotting down. This was actually difficult and looks strange. I'm not sure I got it right, but I thought I was going at it properly. I think not having the width proper or something I don't know. I tried to throw some ellipsis in there to see if it helped, but I don' t know. Anyways, I'm getting plenty of education here already! Cool! It seems I have a slow as hell method, but if I can get it done, I'll be happy for now. I'm sure with time I'll be able to cut some steps I'm doing now. But I like to check alot so thats why I divide most planes up. Crits more then welcome. And thanks for the continued stuff! Name:  Perspecstudy2_27_13.jpg
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    Good stuff, Stray. I just corrected a couple of things that jumped out at me.

    1. For your boat, the constructions all generally look good. The only thing that didn't seem right at first glance (without digging into the construction meticulously) was the cast shadow length. Generally the lower the LVP (light vanishing point) is to the Horizon Line then the larger the shadows will be. You can see this effect happen at dusk/dawn when the sun is low with your own cast shadow. So I've drawn some corrections there for you.

    2. Great job on the construction on this mirrored plane exercise. I started to do this one in a tiny ass SB when I was out of town and just had to stop because of the amount of lines cluttering up my page, ha. The only thing looking weird, as I've drawn out in the corrections, are the ellipses. The most important thing to keep in mind when drawing circles in perspective is the minor axis of your ellipse, since it will follow your perspective lines and dictate how it should look in perspective. In this instance, your RVP is the line which your ellipse's minor axis follows, so we use that line as a guide. The minor axis cuts our ellipses directly in half at the smaller of the 2 diameters (the larger being the major axis), leaving 2 identical halves. The other important thing to note is the degree of the ellipse. The farther out it is, the more "open" the ellipse becomes. I've drawn what this would roughly look like with it corrected. Hope this helps and keep posting!

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    Wow!!! GREAT idea for a thread and some great instruction/exercises here too! Love the idea of making perspective fun by having it a challenge like a puzzle. This is a topic everyone tends to ignore but is very important (something I'm only beginning to realize now, and am regretting not studying it properly in the past).

    I've read through (very quickly) what's been posted so far and it is quite something I'll be back sometime in the near future to see how things progress, and will probably do some of the exercises myself as this is a very weak area in my skillset (and I'll actually sit down and go through everything fully instead of a quick skim).

    Anyhow, for now just thought I'd leave a little comment and tell everyone to PLEASE CONTINUE with this thread! I've rated it five stars, so hopefully if some other people do too it will start to garner a bit more attention



    Oh, and also I noticed that Mr Frenik mentioned using a wide angle lens...maybe in one of your future explanations you can go into the difference between lenses, how things appear in them etc. I don't know much about it, but would be interested in learning. I assume it has to do with how far away vanishing points are? How does it relate to field of view/cone of vision (which I also don't know much about haha)? I also know that different lenses make objects of varying distances appear closer or farther away/bigger or smaller...why is this?

    Last edited by Andrew Sonea; February 28th, 2013 at 05:29 AM.
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    Thanks for the kind comments and rating, Andrew. I'll go over lens differences and how those relate to FOV as well as explain COV. And I appreciate getting this thread out there; you're right that this is stuff that we all tend to put off but is really the most important to learn, as far as technical skill goes. It just got to the point where instead of constantly saying it, I decided to start a thread where we could all study it. So thanks again for helping out and hope to see you in here again soon!

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    Frenik Thanks for the Engineering stuff thats Awesome something i been looking for


    Edit: just one thing tho, im not sure why the front and side views are in the picture plane? ive seen it done another way for architectural perspective, and they keep it above with the top view. i dunno i try find later might be something to think about?)

    Last edited by Siphonophores; February 28th, 2013 at 06:25 AM.
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  47. #24
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    Quote Originally Posted by Care View Post
    Frenik Thanks for the Engineering stuff thats Awesome something i been looking for


    Edit: just one thing tho, im not sure why the front and side views are in the picture plane? ive seen it done another way for architectural perspective, and they keep it above with the top view. i dunno i try find later might be something to think about?)

    I've tried looking around for what you're describing, but what I'm finding is what I've been taught, which is that your front/side views are used as elevations and top view as plans, therefore you must put the side/front views on the ground line in order to get your height information. If you can find something that shows it differently, I'd be interested in checking it out.

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    Thanks for the crits on those last exercises MrFrenik! I had a feeling I wouldn't have gotten the shadows right. I did try but not having them all fit in the image did occur to me as a failure in exercise. I'm mostly glad there were no errors in boat construction jumping right out at you. Also thanks for the axis description, I've only had a brief read over on this in book, and your description is good, but I think it's something I'll really need to practice a ton to understand properly. Mostly construction of ellipses in general is hard for me via freehand or cube construction. It's a shape hard for me to wrap my head around for some reason. PRACTICE ey?

    I also look forward to the new exercises. Clearly valuable stuff. And in agreement with Andrew Sonea above. If you have info on cameras. I would love stuff on this. I see Feng talk alot about different lenses in relation to real camera's.(not sure if it's entirely perspective based all the time) I've also seen some video's where his perspective is warp tooled afterward to reflect fish eye. None of this is covered in Norling, likely fish eye didn't exist then, lolz. But are there specific rules to this? Thanks man, so much.

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    Cone of Vision and Wide Angled Lenses

    This will act as a basic overview of these two topics, mainly because I'm no authority on optics and can only speak about the basic effects of how Wide Angled Lenses work with regards to perspective drawing. Hopefully this stuff helps clear some stuff up, and if anyone more advanced would like to chime in, it'd be much appreciated!

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    So I think we can all agree that doing ellipses digitally is a bit of a pain, especially when you're trying to get a "hand drawn" look about them, much like you can get in SBpro with its ellipse template. I took some time and came up with this template, which uses an inverse mask to allow you to freehand draw ellipses. If anyone knows how to upload the PSD file ( I keep getting errors when trying), I'd be happy to share it with you guys.

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    Thats funny I just studied the hell out of some station point today. And then you posted, cool! I have hard time with cone for some reason, mainly because handprint is super technical. but found the station point by setting horizon, setting ground line at a measurment then multiplying it by 1.73? Is that correct, atleast in one point perspective? It's amazing that also in both Loomis and Norling, perspective depth isn't really discussed, and until questioning it, I always thought the rear of a box was arbitrary, or eyeballed. But I wanted to find out just how to do it, so I learned of Measuring points, found from 45's off of station point. Anyways, any crit on this would be appreciated, and thanks for the follow ups MrFrenik. I haven't any idea about .psd's. Maybe a link in your sig?

    Anyways, need to keep practicing. I think I'm making ground on it, this perspective. Name:  dutchbottle_perspectiveset.jpg
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Size:  65.0 KBBut it's funny how you mentioned having done an exercise on paper and having so many lines. There's no way in my sanity I could manage anyone of these on paper. No way.

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  55. #29
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    Thanks for that post on cone of vision Mr Frenik! Hopefully not a dumb question, but: if the box you have in the diagram is rotated...how does that change things? Like would the vanishing points just move? They can be anywhere?


    I tried out the first exercise which proved a bit harder than I expected given how simple it is. I did it from the view opposite how it was originally drawn to add a challengeand tried to not look at the original at all while doing it. It's not very accurate because I didn't actually plot out the horizon lines or vanishing points or anything and just eyeballed it all haha! And rather than use the "diagonal through the center" to find the points of the curve on the far side, I just eyeballed about haw far the line should end from the farthest line (if that makes sense in writing).

    I was also wondering, how would one ensure that line A and line B (sorry for making it a purple that is very close to the red) are the exact same length? I just drew it so it looks somewhat right...would I need to do the whole method used in the orthographic projection, or is there a quicker way?

    And off topic, but how does one construct shadows for spheres? It's not often that I get to ask questions like this to people who actually know perspective, so I'll take advantage of it haha!

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    Quote Originally Posted by I STRaY I View Post
    Thats funny I just studied the hell out of some station point today. And then you posted, cool! I have hard time with cone for some reason, mainly because handprint is super technical. but found the station point by setting horizon, setting ground line at a measurment then multiplying it by 1.73? Is that correct, atleast in one point perspective? It's amazing that also in both Loomis and Norling, perspective depth isn't really discussed, and until questioning it, I always thought the rear of a box was arbitrary, or eyeballed. But I wanted to find out just how to do it, so I learned of Measuring points, found from 45's off of station point. Anyways, any crit on this would be appreciated, and thanks for the follow ups MrFrenik.

    Finding your station point is relatively simple, actually, so no multiplying is necessary at all. And it seems that you're trying to use trigonometric ratios in order to derive where to place your VP's according to the COV, but again, this isn't necessary. From a plan view (or looking top-down to your scene), imagine where you'd like for your camera to be standing in relation to your object. Remember that normal COV is 60 degrees, so if your object fits within this, then you'll see all of it. If it doesn't, because you're much too close to the object to see the full extent of it, then it will extend beyond your COV and thus out of your drawing. Of course, you could always simulate a wide angled lens and "capture" more of your object. Check this page out https://wiki.umn.edu/pub/DF2TA/Tutor...e-Tutorial.pdf It should clarify a bit more. Let me know if this needs more explanation though.

    And yes, Measuring Points are necessary for measured perspective drawing, something which I plan to cover very soon.

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