# From artwork to toolpath

October 25, 2026 · https://tatbot.ai/updates/2026-10-25/

For his 2020 work **Aligned Movement**, Tyler Hobbs began by drawing a curve in a custom software interface. An algorithm used it to organize strings of dots and suggest their colors. A plotter transferred the dots to paper in pencil. Hobbs then painted them in gouache, following some of the suggested colors and changing others. His [account of the work](https://www.shop.tylerxhobbs.com/works/2020/aligned-movement) describes artistic decisions continuing through the program, the machine and the painted surface.

That continuity is easy to miss when a drawing appears to become a machine instruction at the press of a button. The button can conceal decisions about the meaning of a line, the size of the work, the path of a tool and the pace of its movement. Some decisions are made by the artist. Others arrive as software defaults, material settings or assumptions built into a machine.

A **toolpath** describes the route a tool should take relative to the work, often with an orientation as well as a position. Turning artwork into that route requires choosing which properties to preserve: a silhouette, a centerline, a spacing, a gesture, a relationship between marks. Turning the route into a physical result adds time, calibration, control and inspection. The history of this translation is partly a history of who makes each choice.

## What the image leaves open

A raster image describes a grid of samples. A dark stripe occupies an area in that grid; it does not necessarily record how a hand or tool moved to create it. A vector description can retain lines, curves and relationships between shapes. Those are useful geometric commitments, but they still need an interpretation for the intended process.

Tracing a thick stripe's boundary produces an outline around the stripe. Following its centerline produces a different route. Neither operation automatically reconstructs the original gesture. Peter Selinger's [2003 Potrace paper](https://potrace.sourceforge.net/potrace.pdf), for example, describes converting black-and-white boundaries into polygons and then curves. Its corner examples show why smoothing involves a judgment: retaining too many corners makes one kind of error, while retaining too few rounds away features. The paper also distinguishes corner detection from later optimization, which joins eligible curved segments while preserving detected corners. Fewer segments can retain the chosen features. A small irregularity might be scanning noise, or it might be [the part an artist cares about](https://www.tylerxhobbs.com/words/connected-for-the-moment).

Consider an original, idealized example: a black cross made from opaque strokes of equal width, with round ends and joins. One construction uses a horizontal stroke and a vertical stroke. Another uses two strokes that each turn at the center, one from left to top and another from bottom to right. Both cover the same area. Reversing the order also leaves the image unchanged. With no transparency, pressure or layering cues, the finished image cannot identify those different groupings and orders. A centerline graph can preserve the cross while leaving the traversal undecided.

Even an existing vector drawing needs its units and transformations understood. SVG, a common vector image format, maps drawing coordinates into a displayed region; its viewBox sets the coordinate rectangle for that mapping. A coordinate value alone does not say how many millimeters a machine should travel. Its [coordinate specification](https://www.w3.org/TR/2026/CR-SVG2-20261006/coords.html) also makes the familiar screen convention explicit: the initial vertical axis points down. A physical drawing convention may point up. A reflection or a mistaken scale can survive every subsequent calculation perfectly.

Appearance can hide another distinction. SVG [fills an open subpath](https://www.w3.org/TR/2026/CR-SVG2-20261006/painting.html) as though a closing segment connected its ends. A visibly filled region therefore need not have an explicitly closed path. Fill rules determine which areas count as inside; they do not choose how a pen should hatch those areas or how a cutter should reach their boundaries.

These gaps do not make a vector file inadequate user input. A familiar plotter workflow can supply units, a fitted pen, paper placement and established defaults. The context completes the instruction. Trouble begins when that context changes while the drawing is treated as though it already contains every decision.

## Who does the copying

Mechanical copying makes the division of labor unusually visible. In **Pantographice**, published in 1631, Christoph Scheiner described arrangements of a fixed point, a tracing point and a writing point connected through a linkage. Their positions and proportions determine enlargement or reduction. The person still chooses what to trace and how to guide the tracer; the mechanism enforces a geometric relationship between the two movements. The [original volume](https://www.e-rara.ch/zut/content/titleinfo/265604), particularly its discussion and diagrams on printed pages 24-25, documents this arrangement.

Numerical control moved a different part of the work into a representation. MIT's [1952 President's Report](https://web.mit.edu/src/pres-rep/49-58__Killian/1952.pdf), on printed page 53, reports a completed servo-controlled milling prototype directed by information on punched tape. Motion could be specified numerically and replayed without a person continuously tracing a model. Preparing those numbers remained a substantial task.

A joint MIT and Aircraft Industries Association [explanation of APT dated February 26, 1959](https://cdn.libraries.mit.edu/dissemination/diponline/AC0069_NewReleases/NewsRelease_1950/AC0069_1959/AC0069_195902_002.pdf) described an automatic machine-tool programming system that still relied on a designer, a draftsman and a part programmer. The programmer described a machining sequence; the computer expanded the description into detailed machine instructions. The release's sample language included geometry, tool-side choices, coolant and spindle instructions. Douglas T. Ross's [technical account](https://www.bitsavers.org/pdf/mit/whirlwind/apt/APT_System_Volume_1_General_Description_of_the_APT_System_1959.pdf) describes separate tool-center calculations and output programs tailored to particular machine controllers. The APT system automated calculations and translation while relying on supplied process knowledge. The release itself acknowledged earlier automatic programming systems.

Computer graphics also changed what a drawing could preserve. Ivan Sutherland's 1963 **Sketchpad** thesis described constructing a hexagon through relationships and constraints, rather than treating the picture only as unrelated points. It also documented a plotter drawing lines and circles under TX-2 computer control or from tape. In the [archival edition](https://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-574.pdf), Figure 1.3 shows that output equipment; the coordinate discussion explains scaling to the plotter. The design relationships helped describe the intended shape. A manufacturing process still needed its own decisions.

Commercial plotting made further layers concrete. The December 1982 [HP Journal account of the 7470A](https://hparchive.com/Journals/HPJ-1982-12.pdf) separates interpretation of HP-GL graphics commands from servo-vector generation, pen lifting, motor drivers and optical-encoder feedback. A line command passed through several kinds of representation before paper and pen moved.

These examples overlap rather than forming a single chain of descent. Copying a contour, defining a constrained shape, calculating a machining sequence and controlling a pen solve different problems. Their shared contribution is a way to carry selected relationships from an intention into an action.

## The tool changes the route

The center of a cutting tool usually follows a different route from the edge left on a part. A cutter removes a region around its center; placing that center directly on the desired boundary can remove material that should remain. Tool-radius compensation accounts for that separation. The [LinuxCNC 2.9 documentation](https://linuxcnc.org/docs/2.9/html/gcode/tool-compensation.html) illustrates why corner context also matters: an inside corner and an outside corner require different compensated endpoints. The approach to a contour is part of the problem, too. In flat profile cutting, a circular tool also cannot reproduce a sharp inside corner just by shifting its center, as illustrated in the [NIST interpreter report](https://nvlpubs.nist.gov/nistpubs/Legacy/IR/nistir6556.pdf).

Compensation can be [calculated in CAM](https://help.autodesk.com/cloudhelp/ENU/Fusion-CAM/files/GUID89C78618-20FD-4A82-BFB4-BC66AFE417DD.htm), the software that plans a manufacturing process, or supplied through a controller's interpreter. These responsibilities do not require one software package per stage. What matters is knowing whether a curve describes the desired part edge or a tool-center route, and which stage supplies the offset.

A pen deposits a mark around its contact path. A laser supplies energy to a region of material. An extrusion nozzle lays down a bead. Sending their centers around the same outline does not make these processes equivalent. A filled shape might call for hatching with a pen, scanning with a laser or multiple adjacent extrusion paths. Spacing, overlap and passes become part of the interpretation of the artwork.

Ordering is also a process decision. LightBurn's [optimization documentation](https://docs.lightburnsoftware.com/2.2/Reference/OptimizationSettings/#cut-inner-shapes-first) explains the reason for cutting inner shapes before a closed outer boundary: the outer cut can release a part so that it drops or shifts. A route that saves travel but releases the work too soon fails an objective that distance alone cannot express.

There is measured evidence that sequence changes more than travel. In a [2007 laser-cutting study](https://www.jstage.jst.go.jp/article/jamdsm/1/3/1_3_368/_article), Masayuki Nunobiki and colleagues compared surface profiles before and after cutting slots in clamped stainless-steel coupons. In their 1 mm sheets, inward and outward sequences produced bending in different directions. In 2 mm sheets, all tested sequences bent upward. This was a particular material and setup, not a universal recipe for the best order. It shows why a planner needs a process objective alongside its geometric route.

Material-extrusion printing exposes another tradeoff. Cross-sections must be covered by beads with physical widths, rather than by widthless mathematical lines. Tim Kuipers and colleagues' [2020 adaptive-width study](https://arxiv.org/abs/2004.13497v1) varied bead widths to fill regions that fit poorly into a fixed-width pattern. Their Ultimaker S5 experiments with PLA used a tuned model of material flow to choose movement speeds for the requested bead widths. In the printed comparison, the authors reported more homogeneous filling but less accurate outlines where the number of adjacent beads changed. The paper's large coverage-error comparison was calculated geometry, and its time estimates used firmware simulation. The physical prints, calculated coverage and estimated runtime answered different questions.

For an artist, choosing an objective can be as consequential as choosing an algorithm. A widening option in the printing framework can prescribe a bead wider than a thin modeled feature. Accepting that heavier feature is a different objective from preserving its original boundary. Process planning makes those priorities operational; it cannot discover them from a silhouette alone.

## A curve needs a clock

An original mathematical illustration makes the next translation visible. Take a cubic Bézier curve with endpoints (0,0) and (24,0), and control points (0,16) and (4,16), all in millimeters. The two middle control points shape the bend without sitting on the curve, which fits a rectangle 24 mm wide and 12 mm high. Its calculated length is about 37.26 mm. This is synthetic geometry, not a measured drawing or a reconstruction of Hobbs's work.

The curve has a parameter that runs from zero to one. Dividing that range into twenty equal steps gives unequal distances along the curve: about 0.91 mm in the shortest interval and 3.67 mm in the longest. Giving each step the same amount of time would therefore make the tool speed vary. Dividing by arc length instead gives twenty curved intervals of about 1.86 mm each. The parameter describes progression through a formula; arc length describes distance along its result.

Time scaling assigns a clock to that progression. At an ideal constant 8 mm/s, this curve would take about 4.66 seconds. This nominal travel time leaves acceleration into the curve, stopping, pen engagement and machine limits unassessed. It is not a measured drawing time or a verified machine trajectory. The distinction between a path and a trajectory indexed by time is developed in Lynch and Park's [Modern Robotics explanation](https://modernrobotics.northwestern.edu/nu-gm-book-resource/9-1-and-9-2-point-to-point-trajectories-part-1-of-2/).

Even at constant speed, following a smooth bend requires acceleration because the direction of motion changes. An instantaneous turn at nonzero speed would require an instantaneous change in velocity. A planner must accommodate acceleration limits: it can slow down, stop at a corner or replace it with an allowed blend. Those choices can change the character of a drawn corner. Jerk, the rate at which acceleration changes, supplies another constraint on transitions; [different time scalings](https://modernrobotics.northwestern.edu/nu-gm-book-resource/9-1-and-9-2-point-to-point-trajectories-part-2-of-2/) handle these boundary conditions differently.

The [LinuxCNC planning discussion](https://linuxcnc.org/docs/2.9/html/user/user-concepts.html) explains why a requested feed rate may not be reached on short moves. Its [documented modes](https://linuxcnc.org/docs/2.9/html/gcode/g-code.html#gcode:g61) distinguish visiting a point from stopping there: exact-path mode can continue through collinear moves, while exact-stop mode stops at each segment end. Extra points can therefore add repeated acceleration cycles under one policy without requiring them under another. A planner can also merge nearly collinear moves within a chosen tolerance. [Look-ahead planning](https://linuxcnc.org/docs/2.9/html/config/ini-config.html#sub:ini:sec:traj) uses upcoming moves to choose a speed that leaves room for the next change of direction. The goal is to retain intended features while managing motion, rather than maximizing point count or smoothness.

Curve approximation adds a separate tolerance. Replacing a curve with many short straight segments can keep the route within a chosen geometric distance of the curve. That bound does not establish how closely the machine follows the route or how the material responds. More coordinate digits and smaller segments refine a description; the physical result still depends on the rest of the system.

## Putting the drawing on an object

The same curve can be wrapped around an ideal cylinder of radius 20 mm, measuring its horizontal coordinate around the circumference and leaving its vertical coordinate along the cylinder's axis. This construction preserves local surface lengths. In a front-on projection with parallel viewing lines, its 24 mm circumferential span is only about 22.59 mm wide. A design can preserve distance on a surface while changing its appearance from a chosen viewpoint.

That is a special, explicitly defined mapping. Mapping to a general curved object can stretch or compress the drawing, and preserving angles is a different goal from preserving lengths. A designer must choose which surface relationships or projected appearance matter. Joining successive samples of our curve by straight lines makes chords that pass inside the cylinder. A collection of mapped points and a motion that stays on the surface are different geometric descriptions.

A point also leaves orientation open. The surface normal, a direction perpendicular to the surface, can help define how a tool should approach. It does not determine rotation about that direction or account for the tool's own geometry. Together, position and orientation define a tool's pose. The chosen pose must also suit the tool's contact with the surface.

An articulated arm adds joint motion. A [straight movement through joint angles](https://modernrobotics.northwestern.edu/nu-gm-book-resource/9-1-and-9-2-point-to-point-trajectories-part-1-of-2/) does not generally make a straight tool-tip path. Inverse kinematics finds joint configurations for a desired tool pose; the [route between configurations](https://modernrobotics.northwestern.edu/nu-gm-book-resource/10-1-overview-of-motion-planning/) must also respect joint limits and obstacles. At a singularity, the arm can lose an instantaneous direction of motion, as illustrated in [Modern Robotics' singularity discussion](https://modernrobotics.northwestern.edu/nu-gm-book-resource/5-3-singularities/). A visually smooth curve says little by itself about these joint-space constraints.

A 2020 [robotic drawing study by Ruishuang Liu and colleagues](https://arxiv.org/abs/2009.14501v2) shows what execution adds. Their robot grasped pens in a two-finger hand, observed the pen pose and planned drawing on 3D surfaces. In a star-drawing comparison with a displaced surface model, one tip was missing without impedance control, which lets the tool respond compliantly to contact. With that control, the tip appeared. Correcting contact changed the mark without changing the intended star.

The same study also reported recovery that skipped failed points. A completed run could therefore leave some specified points undrawn. Its geometric mapping comparisons and robot-reported position data were not independent measurements of the pen's physical mark. These distinctions matter when evaluating a system: finding a path, executing it and producing the intended drawing are separate achievements.

## What the mark can tell us

Execution needs a relationship between the drawing's coordinates, the machine and the actual work. For an articulated arm, [calibration can estimate model parameters](https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=933222) that relate joint readings to tool pose. Registration determines the transformation between coordinate frames, so a point in an object model can be expressed in the machine's frame. If paper shifts after alignment, a tool may still follow its commanded route closely while putting the drawing in the wrong place. A controller can use measurements and a model to estimate quantities it does not sense directly. Motor feedback alone may leave that paper shift undetected.

G-code is one way to express moves and process state, including requested feed, in a machine program. It still needs interpretation and motion generation. The [NIST RS274NGC interpreter report](https://nvlpubs.nist.gov/nistpubs/Legacy/IR/nistir6556.pdf) separates the language from canonical machining functions and describes coordinate and tool-offset conventions. A timed trajectory expresses desired motion over time; a feedback controller compares observations with that demand and produces actuator commands. These responsibilities can be combined or revisited in a real system, but they answer different questions.

The physical mark remains important evidence. The HP 7470A's [mechanical designers](https://hparchive.com/Journals/HPJ-1982-12.pdf) reported early pens skipping over paper roughness. Their solution included changing the pen suspension's interaction with the paper. Digital interpretation and encoder feedback did not make contact mechanics disappear. A continuous geometric route could still leave a broken ink line.

In the same 1982 issue, [an account of the HP-41C calculator's plotting module](https://hparchive.com/Journals/HPJ-1982-12.pdf) described connecting the tops of bars within each barcode row to reduce pen drops and nib wear. Settings that worked with a new pen could later produce unreadable code as the nib changed.

Inspection should match the intention. In curve preparation, a geometric tolerance bounds the allowed deviation between a curve and its approximation. Following error compares [commanded motion with feedback](https://linuxcnc.org/docs/2.9/html/config/ini-config.html#sub:ini:sec:joint-num), which may observe a motor or joint rather than the mark. Dimensional accuracy concerns the resulting work relative to its intended dimensions. Visual quality concerns the appearance that motivated the work. A smooth line can be misplaced; a dimensionally faithful outline can have an unwanted line weight. Evidence for one property does not establish all the others.

[Hobbs's plotted dots](https://www.shop.tylerxhobbs.com/works/2020/aligned-movement) left room for another decision. He could follow the algorithm's suggested colors or change them as he painted. A useful artwork-to-toolpath workflow makes such choices explicit: which relationships to carry forward, which physical limits to accommodate, and which changes should send the work back for another decision.