Constrained Geometry
Constrained geometry lets you create lines, arcs, and circles that are tangent to existing sketch elements. Instead of calculating positions and angles by hand, you describe the relationship you want and FluidCAD solves the geometry for you.
The three constrained primitives are:
tLine()— a line tangent to one or two objectstArc()— an arc tangent to objects, points, or the previous elementtCircle()— a circle tangent to two objects
Constraint qualifiers
When a constrained shape is tangent to a circle or curve, there are often multiple valid solutions. For example, a circle tangent to two other circles can sit between them, wrap around them, or fit inside them. Qualifiers let you specify which geometric relationship you want.
import { outside, enclosing, enclosed } from 'fluidcad/constraints';
Under the hood, these qualifiers correspond to the position of the solution relative to the "material side" of a curve — the interior of a circle (the filled region) or the left side of a line (relative to its direction). The three qualifiers describe three distinct spatial relationships:
outside(obj) — external tangency
The solution and obj are external to one another — they touch but do not overlap. For circles, this means the solution sits on the outside of obj, tangent at one point with no shared interior.
Use outside() when you want the solution to stay clear of the wrapped object — like a belt running around a pulley, or a tangent line that does not cross into the circle.
enclosing(obj) — the solution wraps around the object
The solution encloses obj — the argument circle is inside the solution. The solution is the larger shape that contains obj.
Use enclosing() when you want the solution to wrap around the constraint object. For example, a tangent arc that sweeps around a guide circle from the outside.
enclosed(obj) — the solution fits inside the object
The solution is enclosed by obj — the solution sits inside the argument circle. This is the inverse of enclosing(): the solution is the smaller shape contained by obj.
Use enclosed() when you need a solution that fits within an existing circle — for instance, an internal tangent circle nestled inside a larger construction circle.
How qualifiers reduce the number of solutions
Without any qualifier, FluidCAD returns all geometrically valid solutions. For a circle tangent to two other circles, there can be up to 8 solutions. Qualifiers act as filters — each one constrains the spatial relationship for that argument, which narrows the result set. Combining qualifiers on both arguments (e.g., outside(c1) and enclosing(c2)) narrows the results further, often to just one or two solutions.
| Qualifier combination | Meaning |
|---|---|
outside(c1), outside(c2) | Solution is external to both circles |
enclosing(c1), enclosing(c2) | Solution wraps around both circles |
enclosed(c1), enclosed(c2) | Solution fits inside both circles |
enclosing(c1), outside(c2) | Solution wraps around c1 but stays external to c2 |
| No qualifiers | All valid solutions are returned |
All solutions (no qualifier)
With two guide circles and no qualifier, tLine() returns every valid tangent line:
import { circle, sketch } from "fluidcad/core";
import { tLine } from "fluidcad/core";
sketch("xy", () => {
const c1 = circle(100).guide()
const c2 = circle([200, 0], 40).guide()
tLine(c1, c2)
})
Filter with qualifiers
Each qualifier applies to the individual object it wraps. outside(c1) keeps solutions that are tangent to c1 from the outside, while enclosing(c2) keeps solutions that enclose c2. Combining qualifiers on both arguments narrows the results to just the lines matching both constraints:
import { enclosing, outside } from "fluidcad/constraints";
import { circle, sketch } from "fluidcad/core";
import { tLine } from "fluidcad/core";
sketch("xy", () => {
const c1 = circle(100).guide()
const c2 = circle([200, 0], 40).guide()
tLine(outside(c1), outside(c2))
tLine(enclosing(c1), enclosing(c2))
})
The mustTouch option
tArc() and tCircle() accept an optional mustTouch boolean as their last argument. When true, only solutions that physically touch both objects are included:
tArc(c1, c2, 50, true) // only arcs that touch both c1 and c2
tCircle(l1, l2, 200, true) // only circles that touch both lines
This is useful when the solver finds multiple valid tangent arcs or circles and you only want the ones that are in contact with both inputs.
tLine
Tangent continuation
The simplest use — draw a line that continues in the current tangent direction (shown by the orange arrow in the viewport):
sketch("xy", () => {
arc(50, 0, 90)
tLine(100) // 100 units in the tangent direction
})
Tangent to two objects
Draw a line tangent to two circles, arcs, or curves:
import { outside, enclosing } from 'fluidcad/constraints';
sketch("xy", () => {
const c1 = circle(100).guide()
const c2 = circle([200, 0], 60).guide()
tLine(outside(c1), outside(c2)) // tangent on the outside of both
})
Different qualifiers give different tangent lines:
tLine(outside(c1), outside(c2)) // crosses between the circles
tLine(enclosing(c1), enclosing(c2)) // wraps around both circles
Tangent between arcs
sketch("xz", () => {
move([-20, 0])
const a1 = arc(100, 0, 180)
move([50, -150])
const a2 = arc(50, 270, 0)
tLine(a1, a2) // line tangent to both arcs
})
Accessing tangent line endpoints
The result of a two-object tLine() exposes .start() and .end() vertices, which you can use to connect other geometry:
const t1 = tLine(outside(c1), outside(c2))
const t2 = tLine(enclosing(c1), enclosing(c2))
// Connect the tangent lines with arcs
tArc(t1.end(), t2.end(), t1.tangent())
move(t1.start())
tArc(t2.start(), t1.start(), t1.tangent().reverse())
tArc
tArc() is the most flexible constrained primitive. It can create arcs in several ways.
Tangent continuation
Continue from the current position and tangent direction:
sketch("front", () => {
vLine(100)
tArc(50, 180) // radius 50, sweep 180°
tArc(80, -270) // radius 80, sweep -270° (clockwise)
})
Negative angles sweep clockwise, positive angles sweep counter-clockwise.
Tangent to two objects
Draw an arc tangent to two circles, lines, arcs, or points:
sketch("xy", () => {
const c1 = circle(160).guide()
const c2 = circle([200, 0], 60).guide()
tArc(outside(c1), outside(c2), 80) // radius 80, outside both
})
Between a circle and a line
sketch("xy", () => {
const l = aLine(150, 45)
const c = circle([100, 0], 40)
tArc(c, l, 50).guide() // radius 50, tangent to both
})
Between two lines
sketch("xy", () => {
const l1 = aLine(150, 45)
move([-50, 0])
const l2 = vLine(100)
tArc(l1, l2, 50).guide() // fillet-like arc between two lines
})
Through two points
sketch("xy", () => {
tArc([-50, 0], [50, 0], 150) // arc through two points, radius 150
})
From object to point
sketch("xy", () => {
const c = circle([100, 0], 40)
const p = [100, 50]
move(p)
tArc(outside(c), p, 100) // arc from circle to point, radius 100
})
tCircle
tCircle() creates a full circle tangent to two objects. It takes the two objects and a diameter.
Tangent to two circles
import { outside, enclosing } from 'fluidcad/constraints';
sketch("xy", () => {
const c1 = circle(160).guide()
const c2 = circle([200, 0], 60).guide()
tCircle(c1, enclosing(c2), 160).guide() // tangent to c1, enclosing c2
tCircle(outside(c1), outside(c2), 160).guide() // outside both
})
Tangent to two lines
sketch("xy", () => {
const l1 = aLine(300, 45)
move([-50, 0])
const l2 = vLine(300)
tCircle(l1, l2, 200, true).guide() // diameter 200, tangent to both lines
})
The fourth argument is mustTouch — when true, only solutions that touch both lines are returned.
Between a circle and a line
sketch("xy", () => {
const l = aLine(150, 45)
const c = circle([100, 0], 60)
tCircle(c, l, 100).guide()
})
Through two points
sketch("xy", () => {
tCircle([-50, 0], [50, 0], 300) // diameter 300, through both points
})
Common patterns
Using guides for construction
Constrained geometry often works with .guide() shapes — construction circles and lines that define the tangent relationships but aren't part of the final profile:
import { sketch, move } from 'fluidcad/core';
import { circle, tLine, tArc } from 'fluidcad/core';
import { outside, enclosing } from 'fluidcad/constraints';
sketch("xy", () => {
const c1 = circle(100).guide()
const c2 = circle([200, 0], 60).guide()
const t1 = tLine(outside(c1), outside(c2))
const t2 = tLine(enclosing(c1), enclosing(c2))
tArc(t1.end(), t2.end(), t1.tangent())
move(t1.start())
tArc(t2.start(), t1.start(), t1.tangent().reverse())
})
Smooth profiles with tangent chaining
Use tArc() after lines to create smooth transitions. The tangent direction carries forward automatically:
sketch("front", () => {
vLine(100) // straight up
tArc(50, 180) // smooth turn
tArc(80, -270) // another smooth turn
tLine(50) // continue tangent
})
Each element starts where the previous one ended, and tangent arcs/lines maintain G1 continuity (no sharp corners).