Transform

Transform changes the selected shapes by exact amounts: it scales, moves, turns and reflects them, all in one step, about a point you choose. It is the numerical twin of dragging handles on the canvas, and it shows you the result before anything changes.

What it does

The TRANSFORM section sits under ALIGN on the Controls tab while the Move tool is in hand. Select some shapes, set any of these, then press ‘Apply’:

• Scale, ‘Horizontal’ and ‘Vertical’, in percent: 100 is no change, 50 is half size. The chain-link button beside the heading keeps the two the same and is on to begin with; turn it off to stretch in one direction only. The sliders run from 1 to 200 percent, and the number fields take anything from 1 upward.
• Move, ‘Horizontal’ and ‘Vertical’, a distance in the unit shown beside the field. Down is positive, as it is on screen. Each slider reaches the length of the sheet’s longer side either way, and the fields take any number.
• Rotate, an ‘Angle’ in degrees, from −360 to 360. Positive turns counter-clockwise, as mathematicians measure angles. Drag the hand around the dial, or type in the field; Shift while dragging snaps to 15°. With the dial focused, the arrow keys turn it a degree at a time, or 15° with Shift.
• Reflect X mirrors across the horizontal axis, turning shapes upside down. Reflect Y mirrors across the vertical axis, left to right. The names follow the mathematician’s habit: a reflection ‘in the x-axis’ is the one that flips up and down.
• The reference point, the grid of nine squares: the corners, the middles of the sides and the center of the box around the whole selection. Scaling, turning and reflecting happen about that point, and it is the one point that stays still. The center is chosen to begin with.
• Random and Copies, which repeat the transform; they have their own chapter. With copies at 0, Apply changes the selected shapes themselves.

With ‘Preview’ on, as it is to begin with, the canvas shows a cyan outline where each shape will land and a thinner dashed outline where its piece will fold out to. Paper put back has no piece of its own, so it shows only the first. The outlines are guides only: nothing in your design changes until you press ‘Apply’.

Inside a single step the order is fixed: reflect and scale, then turn, both about the reference point, and then move. Because the move comes last, it is always a straight shift by exactly the distance you typed, however far the shape has turned.

Two things surprise people. First, the reference point belongs to the whole selection, not to each shape. Scale three scattered shapes to 50 percent about the center and the spaces between them halve as well, as though the group were one picture. Second, after ‘Apply’ every setting goes back to no change, so the preview does not at once offer to do the same thing again. ‘Reset’ does the same without applying. The whole transform is one step for Undo.

Why it works this way

Hands are good at feel and bad at exactness. The handles on the canvas are for feel; this section is for the moments when a shape must be exactly half the size, or turned exactly a right angle, or moved exactly the width of another. Typed numbers are also repeatable: the same settings on another selection give the same result.

Preview-then-Apply is deliberate. In Illustrator, the program this section follows, the Transform effect stays live, so its copies follow later edits to the original. Here a transform becomes ordinary cuts, each selectable and editable on its own, and the preview is drawn as guides rather than as temporary cuts, so an undo, an autosave or a change of tool can never find a half-finished transform in your design. Seeing the dashed pieces before committing matters more here than in a drawing program, because every cut sends a piece outside the sheet, where it may land on others.

The metaphor

Transform folds several drawing instruments into one panel: the pantograph for enlarging and reducing, the protractor for the angle, the mirror for reflection. The reference point is the pin. Put a drawing pin through a sheet of tracing paper and turn the sheet: everything moves except the point under the pin. The locator lets you choose where the pin goes.

The metaphor breaks at reflection, and the break is instructive. You cannot turn a drawing into its mirror image by sliding or turning it on the table; you have to lift it and turn it over, through the third dimension. That is why a reflection is the odd one out among the transforms – and it is exactly what a fold does, which is why reflection sits at the heart of this whole app.

A history

In art

For most of history, changing the size of a drawing meant a grid. Painters squared up a sketch – ruled it into squares – and copied it square by square onto a larger grid on panel or wall, the same trick the Egyptians had used for their figures. In 1603 the Jesuit astronomer Christoph Scheiner devised the pantograph, a jointed parallelogram of rods that copies a drawing at an adjustable scale as one point traces it; he described it in print in 1631 in a book called Pantographice.

Symmetry is the study of transformations that leave a design looking the same, and the mathematician Hermann Weyl’s Symmetry (1952) is still the most generous introduction to how art, ornament and nature use them. Every setting in this panel is one of his tools.

In printing

The printer lived in a mirrored world. Type is cast wrong-reading so that it prints right-reading, and a woodcut is carved as the mirror of the picture it makes. A compositor read the letters in the stick backwards and upside down as a matter of course.

Scaling, in the studio, was a percentage. Before software, designers sizing artwork for the process camera used a proportional scale, a circular slide rule with the original’s size on one wheel and the reproduction’s on the other, and a window showing the percentage. The percent in the Scale fields comes from that world. So does the locator: page-layout programs such as Adobe InDesign have long shown nine small squares, which InDesign calls the reference point locator, to say which point of an object stays put.

In technology

Every combination of scaling, turning, reflecting and moving is an affine transformation: it keeps straight lines straight and parallel lines parallel. In 1827 the mathematician August Ferdinand Möbius introduced homogeneous coordinates, and computer graphics adopted them for a practical reason. Write a point (x, y) as (x, y, 1), and a move becomes a multiplication by a small grid of numbers, a matrix, just as scaling and turning already were. A whole sequence of transforms then collapses into one matrix. PostScript, the page-description language behind desktop publishing, keeps its current transformation as just six numbers: four for the scaling, turning and reflecting, two for the move.

Matrices have one famous property that explains the fixed order here: they do not commute. Turn then move is not the same as move then turn. Sketchpad, in 1963, could already copy, move, turn and resize parts of a drawing; every drawing program since, including this one, is built on the same small matrices.

In professional software

This is the transform panel of every vector program: typed numbers, a reference point and a preview. What changes from one program to the next is which way a positive angle turns, which way y runs, and whether scattered shapes transform about one shared point or each about its own.

• Adobe Illustrator. The Transform panel, ‘Window › Transform’ (Shift+F8), has the same nine-square locator beside typed position, size and angle. The Scale (S), Rotate (R) and Reflect (O) tools open a dialog with ‘Preview’ and ‘Copy’ when you double-click the tool, or Option-click (Alt-click) the artwork to put the pin there first. ‘Object › Transform › Transform Each’ scales and turns each shape about its own reference point, which this panel does not do.
• Adobe Photoshop. ‘Edit › Free Transform’, ⌘T (Ctrl+T), puts handles around a layer, and the options bar gives the same locator with typed fields for position, size and angle. One trap: a positive angle there turns clockwise, the opposite of Illustrator and of this app.
• Autodesk AutoCAD. MOVE, ROTATE and SCALE are separate commands, and each first asks for a base point, the pin. ROTATE and SCALE offer a Copy option and a Reference option, which turns a known angle to a new angle, or scales a known length to a new length, rather than by an amount you type. Positive angles turn counter-clockwise by default, and y runs up the screen, not down.
• Autodesk Fusion (formerly Fusion 360). Move/Copy (M) moves and turns bodies and sketch geometry, either by dragging its arrows and rings or by typing distances and angles. Scale is a separate command.

What carries over. Choose the pin before the numbers: where a shape ends up depends on the reference point as much as on the amount. Then, whenever you change program, check two conventions, the direction of a positive angle and the direction of y. A value that is right in one program can send a shape the wrong way in the next.

Try this

• Select a shape, type 45 in the angle field, and click the locator’s top-left square, then the center. Watch the preview swing about each.
• Select two shapes far apart and scale them to 50 percent about the center. They move toward each other as well as shrinking.
• With a cut selected, turn on Reflect Y and compare the dashed piece in the preview with the piece you have now.

Going further

Repeat a transform with Copies, or line shapes up with Align. Move’s unit is set under Inches and centimeters.

• Hermann Weyl, Symmetry