Drawing Flowing Curves
Whether we are drawing lines to be cut or to be drawn with the pen, adjoining curves should flow gracefully from one to the next.¬† We have all had the experience of watching how terribly small angles cut.¬† There is either a shallow corner where the blade withdraws, twists, and plunges again - or there is a plowed spot where the blade twists in the matboard while it is on the move.¬† Neither of these cutting scenarios is particularly attractive.¬† At times these small angles are critical to define an image, but more often the transition could be smooth and the slice would look so much nicer.
If we are drawing lines with the pen, so many elements are so small, you would not imagine how a drawing glitch here and there would even be visible.¬† But when curves meet at small angles, the pen instantaneously changes direction and there could be an erratic wiggle in the line.¬† We could call these wiggles ‚Äúcharacter,‚ÄĚ or we could smooth them so that there are no wiggles.
In either case - cutting or drawing - the fix is pretty easy once you understand how the tools work.¬† The examples here use Wizard‚Äôs PathTrace, but every drawing program has an automatic smoothing function of some description, though the specific operation sequence may differ.
The Tangification Function
Two arcs join each other.¬† In this illustration, one is gray and the other is white.¬† Look closely at their junction point.¬† They meet at a slight angle.¬† If this is a drawing to be cut, this is the point where the blade may withdraw, shift, and plunge again.¬† Or the blade may instantaneously change twist as the machine keeps moving.¬† This angle can be smoothed using the Tangification function.
Here is how Tangification operates.¬† Click the Prep Design tab at the top and select Join Segments under Choose Operation at the left.¬† First join the two arcs.
Next, select Tangify Segments under Choose Operation.¬† At the bottom left is the Maximum Angle field.¬† You can enter any value and choose the maximum allowable angle here.¬† Notice that it is set to 22¬į here.¬† This means that any angle above that will be left alone.¬† Any angle less than 22¬į will be smoothed.¬† The arrow in the drawing field points to the junction of the two arcs.
Move the cursor to highlight the joined arcs and click one time.¬† The transition will be smoothed and the two arcs will now join at a 0¬į angle.¬† Again, the arrow in the drawing field points to the junction of the arcs.¬† After tangification, the angle at the junction is now smoothed.
If the tangified curves were superimposed onto the original curves, you could readily see that tangification works by altering portions of both arcs where they meet.¬† Most of the change here is seen in the larger arc on the right.
There are times when the particular changes made by the tangification function alter the curves in a disagreeable way. ¬†There is a way to take advantage of the tangification function and minimize its impact on the design.¬† We will do this by cutting away pieces of the original arcs and inserting a shorter curve between the larger curves.
Inserting a Transitional Curve
Here is an example where we could not accept the changes to the curves that tangification might perform as it smoothes the transitions.¬† There are adjacent curves to be considered for both spacing issues and aesthetic appeal.
The lines in gray are the original pen ornament.¬† The swirls of the ornament are only a little more than a quarter inch high and the lines that cross in the center are not even a sixteenth of an inch apart.
We want to make the larger flourish on the left a little more grandiose to suit another picture more proportionately.¬† The white curves represent our proposed changes.
There are three elements to the new white curve.¬† There is one curved line at the top and another at the bottom.¬† At the left there is an arc.¬† The segments are joined in this step, so they appear as one curve.
Arcs are very handsome when you are able to incorporate them, because of their constant rate of curvature.¬† Curved lines (when you move the midpoint of a line, it turns into a curve) are easy to use because you are able to move the midpoint from one end of the line to the other.¬† As you do this, the rate of curvature is greater on the shorter end of the line and you are able to blend adjoining elements nicely.
These three elements have been joined and tangified and everything looks just as it should except there is a slight disagreeable flat spot on the curve near the arrow.
There is a Move Point function in PathTrace that may help.¬† Click the Trace and Draw tab at the top.¬† Select Move Point under Choose Operation at the left.
The illustration shows that Move Point is not always the answer, though.¬† After tangification, there are more control points along the curve.¬† The curve sometimes behaves more like a wet noodle than a curve as you move the control points.
The illustration also shows that the action of Move Point is only for discrete elements, joined or not.¬† The angle you see to the right of the arrow is where the curved line at the bottom joins the arc on the left.¬† Patience and experience with the Move Point function could likely correct this flat spot, but inserting a transitional curve is sure to leave all the other portions of the curve undisturbed.
First, we need to decide where to break the curve.¬† Click on Draw Line under Choose Operation.¬† Draw lines approximately equidistant on either side of the flat spot.
The exact position where these lines intersect the curve is not critical. ¬†As you place the lines, imagine the transitional curve and how long it needs to be in order to improve the curve.
Click the Prep Design at the top.¬† Select Break Intersection under Choose Operation.
Break the intersections that the lines (the lines we drew in the previous step) have with the curve.
In this illustration, the white segment is the piece that will be replaced.¬† In this zoomed-in view, you can see that it is less than graceful.
Delete this line.¬† It may possibly be helpful as a reference, but more often, it is in the way of your view as you set to improve the curves.
We will now connect the pieces of the broken curve with a line segment.¬† We could certainly draw a new line, but we have four broken line segments already snapped to the broken endpoints of the curve.
Click the Trace and Draw tab and select Move Point.¬† Move the endpoint of one of the broken line segments on the left and snap it to the broken end of the curve on the right.
The white line in this illustration is the moved line segment.
With the Move Point function still selected, move the midpoint of this intervening line.¬† As you move the point, the line will curve.¬† This is a small segment and tangification will fix nearly any imprecision at this point, but watch how the curve changes as you move the point left and right as well as up and down.¬† With a little experimentation, you will be able to bend the line so that it connects the broken pieces of the curve without a hiccup.
In this illustration the white curve makes the transition in the broken spot of the curve.¬† We have not disturbed the other portions of the new curve that we like so much.
Join all the new segments and tangify them.¬† Unless there are grievous errors, you will not see a change after you click the Tangify Segments button.
In this illustration, all the leftover pieces of the original ornament have been deleted.¬† The new ornament has been given its cutting instructions.¬† It has been set to draw with the pen.¬† The new ornament can now be saved to use over and over.
As you are evaluating this idea for its practicality, you may imagine that all these steps are much more involved than simply refining the curve using the Move Point function, whatever its pitfalls.¬† This may at times be the ideal answer. ¬†But do not discount the big advantage of this alternative procedure. ¬†Inserting a transitional curve to repair a problem area leaves the other areas of the drawing untouched.
Brian Wolf has been a picture framing educator since 1979, specializing in decorative matting techniques. He is the artistry ambassador for Wizard International, Inc. Contact hime at WizardU@wizardint.com ¬†Brian's column is sponsored by Wizard.¬†¬†www.wizardint.com¬†or call 1-888-855-3335