What's A Curve
What’s a curve?
What’s a curve? In fact, what’s a form? Because curves are so readily available to us (just make a circle with your arm, or finger), let’s start with curves.
So, make a circle with your arm, a big circle, from your shoulder, across the front of your body. Now make a smaller one and a smaller one than that. Next, make a circle from your elbow, and then one from your wrist, and now one from your finger. Four different centers from which to generate circles. As you are doing this, imagine that you’re holding a piece of charcoal in your hand as you make these circles, and it is tracing out the forms, permanently, on a large sheet of paper. Choose one of these gestures and now do it again, but jerkily and rhythmically. And again, only this time with little loops in place of the jerks. Or with little z-shaped saw-toothed movements. So far, we have generated a series of circles of varying sizes with smooth to serrated circumferences.
Now, change the speed of the movements; do them fast (throw a fastball) and do them slowly, then very slowly, in excruciatingly slow motion. Next accelerate from a standing start to a whip. And then do them with deceleration (Put on the brakes!). As you do these, imagine that you are holding a fat paintbrush, filled with red paint and it leaves a tracing on the sheet of paper of movement and speed. Change the direction from up to down, or the reverse. Change the orientation. If your circles have been generated across your body (parallel to your chest), try some going at right angles to that, or horizontally, parallel to the floor, like twirling a lasso. In place of the full circles, perform an arc. Do the curves iteratively, over and over, as if you were painting a wall.
Next, let’s try doing these shapes in three-dimensional space, forming spirals, like springs, coils or helixes. Have them expand conically, or contract. Try having the curves start large and get increasingly tighter by first starting from the shoulder, then elbow, then wrist, and finally finger. Now do the arm movement while rotating your body in space.
Is it still a curve if it is made out of a series of short, straight segments? What if there is one bend? What if it is composed of a sequence of still positions in space, like a movie reel? As in old films, how jerky can the movement be for it to hold up as a continuous curve? What if it starts as a planar curve and folds into another plane?
Even a ball dropping down to the ground in a straight line, has a curve embedded in it: the curve that shows its acceleration, and as it bounces, its deceleration, and dampening effect.
The point here is, or begins to be, that whatever we might have started with as an image of “curve” can be seen to be quite limited in contrast with what else can be generated as a series of variations.
And all this, for a dancer, is only the beginner’s manual. There is so much more, more subtlety, richness and complexity available. How does one movement combine with another simultaneous movement? How does one segue into the next? How do you begin, and end?
How do the arm circles combine with a series of pirouettes? What three-dimensional shape do they make in space, as you run? If you were holding a laser, what shape would it make in space; what shape would it trace on the wall?
From the vast catalogue of movement, how do you create a dance, a piece of music, a painting, architecture, a financial recovery plan? What are you trying to do – what is the design intention? Or is it OK to just be fooling around and see what shows up?
And while we are seemingly exploring choreographic movement, gesture and dance, we are also generating spatial examples that can be transformed into other media. What would each of these movements sound like if they were played on a saxophone? Or a piano or guitar? What would they look like as paintings or sculptures? Blown up to a really large scale, how would they appear as architecture? Can you imagine a taste curve, or how a masseuse would translate curves into a body massage?
[ILLUSTRATIONS: Guggenheim Museum, Zaha Hadid, Jackson Pollock, Martha Graham, Edgerton photographs, Eames chair, the Birdcage - Beijing Olympics]
Let’s be clear that none of these physical, graphic, sonic or other gestures in themselves have any inherent meaning, but as creators we often intend for them to be expressive, literally, or emotionally, or to refer abstractly to something else. A sweeping horizontal arm movement is meant to convey all-inclusiveness. A little vertical finger wag says, “Come over here.” Two fingers in a “V” say “peace.” An upraised fist means “power.” A middle finger….
After all, it’s just another form of (non-verbal) language. What is the meaning of an orchestra conductor’s gesticulations to his musicians? He is communicating physically, and symbolically, to indicate musical rhythm, volume, and mood – the abstraction of sounds.
Most of the time, even verbal language is a substitute for experience: the word “green” is not green – it simply is a common sound or, in writing, a visual indicator for the color of most leaves. ( It is actually a frequency range of light that our brain register s as “Sad” is not sad – it indicates a kind of mood or feeling – abstractly, but in a way that we can understand and experience.
Our imagination is limited to what we already know, and we’re satisfied with that. But, as we have seen, there’s no creation in that; repetition, reorganization, rearrangement, perhaps, and there’s nothing wrong there, just no creation. Frank Lloyd Wright, Jackson Pollock, and Martha Graham really invented new forms.
What, then, is a form? What is anything? What can you imagine it to be? What else? You could ask “What else?” a thousand times, and more. How many variations can you come up with? None of them are right or wrong; maybe one is more appropriate, or is, intuitively, to your liking, or expresses an idea best.
How do you know? How can you tell?
Agreements / Ground Rules
Be on Time -
Nothing can be accomplished if this cannot be accomplished first. Be responsible and true to your word on this. Circumstances (we know what they are) get in the way. How can you be victorious in the face of all those circumstances?
If it takes you an hour to get here, door-to-door, leave an hour and 15 minutes early, every day. Show up powerful, ready to work like a champion. Communicate, if you're going to be absent or late, for wherever reason.
Participate -
There are things you won't want to do. If you don't write well, you won't want to write.
If you don't know yourself to draw well, you will resist drawing. If you are habitually late, you won't want to promise to be on time. If you are shy, or easily embarrassed, not full of bravado, you will want to take a day off if you have to present. All of this is manageable if you are willing to give us and each other your word - that you will be on time and execute all of the exercises and assignments that we will do.
We are most comfortable with what we know how to do well. This is not a course about displaying the skills that you already have. This is about an expansion of yourself into the realm of the unknown -THE UNKNOWN. This is really where the good stuff is - creativity, life and the ability to see solutions and beauty where there exists entrenched chaos and hopelessness.
Do not Gossip -
There will be no gossip. This is a promise we will make to each other. What is gossip? That's where you talk about someone in a negative way to someone who cannot make any difference to the subject matter. It is leaving communications anonymously, where they hurt and destroy. If you make the promise, you must also agree not to listen to gossip and to direct anyone who is gossiping to the subject of their discourse. If you have a complaint, take it to one of the faculty only. If you give your promise to this, you will have no trouble knowing if you are gossiping or listening to it - you will feel it and it will feel bad.
These are the underlying values of the studio concept and relationships. You are not here as a single person trying to accomplish a goal. Though that is a worthy thing in itself, this is not that. Here, you rely on each other for your success. And success here is not a linear progression. It is filled with setbacks and studded with breakthroughs.
And this is where we make a promise to you: that you, by agreeing to these three ground rules, will experience a breakthrough, a breakthrough in expressing yourself and in understanding and trusting yourself to design solutions in your environment. You will do great work here - each and every one of you. We promise you that.
Distinctions of Seeing: Principles of Perspective
(Things We Don’t Know We Know)
- “Perspective Drawing: Basic Principles” (Irma Ostroff)
- “Architectural Graphics” (Francis Ching), Chapter 6
- www.khulsey.com/perspective-2pt.html.
Perspective is not magic. It can be learned by following a set of instructions. But it is basically consistent with how we naturally see, a human function that we have taken for granted (have no distinctions around) and are generally unconscious to. The following points are basically invariant principles or laws, like in physics, of how we see.- What you see is determined by where you’re looking from (Station Point). There are an infinite number of station points (around, above, below, closer, farther). If you move your station point, you change your view.
- You see most clearly within about a 60 degree Cone of Vision. Outside of that is peripheral vision.
- The apparent size of things diminishes with distance.
- Lines not parallel to the Picture Plane (where the image is cast) are progressively shorter (Foreshortening) the more perpendicular they are to the Picture Plane.
- Parallel lines converge towards Vanishing Points.
- Hidden Lines exist, but are not visible from the Station Point.
- The size of the image is determined by the position (relative to you at the station point) of the Picture Plane, like a movie screen.
- There is always an (imaginary or real) Height Line, like a surveyor’s measuring rod.
- We see surfaces and tones or color, not lines, but, fundamentally, perspectives are constructed from lines.
- The image is actually re-constructed in your brain as a two-dimensional
Projection, like a slide in a projector.
Perspective Drawing: Basic Principles
This paper has been revised from an earlier draft written by Prof. Irma Ostroff.
All ordinary perspective drawing is based on a simple concept: that between the eye of the observer and the object to be drawn, there stands a transparent plane, a sort of window called the “Picture Plane”, on which the form of the object is projected. This picture plane may be a window in a literal sense, and, if you make the following experiment, the real meaning of the term will be much clearer.
Select a window having large individual panes. Keeping your head as still as possible, using a magic marker, trace on the window the outlines of the things you see through it. (Be sure to stand at the window at a convenient and comfortable distance to mark the window with your arm outstretched.) The result will be a perspective drawing on the picture plane itself.
If you now examine the window, and the scene includes buildings or other objects containing straight lines and right angles within the view, you will notice several important things critical in the practice of drawing perspectives.
- First, all lines appear to be shorter than their true length and the shortening effect increases as the distance of the object increases.
- Second, vertical lines appear truly vertical, whereas horizontal lines, with the exception of lines at eye level, do not appear horizontal.
- Third, groups of parallel horizontal lines, lines running in a single direction, appear to converge towards a single point. Other groups of horizontals having different directions have different points towards which they converge.
- Fourth, these points of convergence for horizontal lines all lie on a horizontal plane, level with the eye of the observer. This effect can be seen most clearly simply by looking down a long straight railway track and noticing that the rails seem to converge on a line emanating directly fro your eye.
You may notice other effects as well. The apparent shortening of a line seen obliquely is called foreshortening: it is more subtle, though often more dramatic, than that of distance, and it exerts a strong effect on the appearance of the object. For example, the apparent flattening of circles results in (slightly distorted) ellipses.
Since vision is impossible without light, it is assumed that a ray of light (line of vision) enters the eye from each (pixel-like) point on the object. On the way from object to eye, each of these rays passes through (pierces) the picture plane. The spot where the ray from any given point on the object pierces the picture plane is the perspective projection of that point. When all the rays from all the (visible) points on the object have produced their perspective projections, the sum total is the perspective projection upon the two-dimensional picture plane of the three-dimensional object beyond.
The “discovery” (distinction) of the principles of perspective was made in the Renaissance by Filippo Brunelleschi. He actually devised a rectangular frame with two cross wires, one horizontal, one vertical, which could be adjusted to intersect at any position within the frame. The intersection then would be positioned so that it corresponded with a particular point on the object that was seen through the frame. Each point on the object had a unique corresponding point in the frame, which was literally, the picture plane. As each point in the frame was identified, it would be transferred to a sheet of paper, thereby constructing, point by point, the two dimensional image of the object. In other words, to make a literal transcription of what you see, simply transfer the projection on the picture plane to paper. Note that this is the transfer of the projection on the picture plane, not the object itself. Interestingly, the computer-aided drawing algorithm for perspectives, engages a digital frame-like device, essentially identical to that which Brunelleschi invented.
The picture plane is usually considered to be perfectly vertical, assuming that you are looking straight ahead. (This is the case in the window drawing described above.) From other points of view (station points), the picture plane may be horizontal (a bird’s eye view) or tilted (looking up at a tall building).
The picture plane may be assumed to be at any given or desired distance from the eye. When it is close to the eye and distant from the object, we see an image small in size, compared to the object; when it is close to the object and distant from the eye, the image approaches the size of the object. In fact, if the picture plane is located beyond the object the resulting image will be larger than the object. Another way to envision this is to imagine that the small image in your eye is projected by a slide or film projector onto a screen. The farther the screen is from your eye (the projector), the larger the image (and the image, aside from its size, is always the same).
In any rectangular object, there are three sets of parallel lines, mutually perpendicular to each other: lines running from top to bottom, lines running from side to side, and lines running from front to back. Each set of lines, actually parallel to the object, tends in the perspective image to converge towards a point in the distance. If there are three such sets of lines, there are consequently three such points. When the picture plane is parallel to any one of these sets of lines, one of the points “disappears” and the lines in question are truly parallel in the perspective image (two-point perspective). Occasionally the picture plane is parallel to two sets of line in the object: in this case, two of the three sets of lines will appear truly parallel in the perspective, and only one point of convergence is needed (one-point perspective).
These points of convergence in perspective images are called vanishing points, and they are fundamental in making perspectives. When the picture plane is vertical, as it is usually, it is naturally parallel to vertical lines in the object, which appear truly vertical in the image, and, naturally, parallel to each other. Moreover, the two vanishing points of the horizontal lines both lie on the same horizontal line, which is also where the station point (where you are looking from) is located. In other words, the vanishing points are on what is called the horizon line which is determined by the height of the station point (eye level). When the picture plane is parallel to both the vertical lines and a set of horizontal lines as well, those horizontals appear as truly horizontal in the image.
When, as often happens, there are sets of lines in an object that are not parallel to the three principle sets, these lines have their own separate vanishing points. Sloping roofs or stairs are examples of this case.
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