Construct an exact circumcircle

Build a ruler-and-compass construction

The mathematical points, lines, and circle retain rational coordinates and construction provenance. Draw is an explicit snapshot step that lowers those values to the same core Graphic understood by SVG and Canvas.

Runnable RiXRun cell

.Plugin.Load("geometry"); a1 := .geometry.Point(0, 0); b1 := .geometry.Point(6, 0); c1 := .geometry.Point(2, 4); abBisector1 := .geometry.PerpendicularBisector(a1, b1); circumcircle1 := .geometry.Circumcircle(a1, b1, c1); center1 := .geometry.Center(circumcircle1); radiusSquared1 := .geometry.RadiusSquared(circumcircle1); construction1 := .geometry.Draw( [abBisector1, circumcircle1, a1, b1, c1], {= view=[-1,-2,7,6], size=[560,560] } ); construction1;

Center and RadiusSquared validate that their argument is a circle before returning its exact defining measurements. This keeps consumers independent of the circle record's storage layout.

Explore a retained construction interactively

The workbench uses the same construction graph for dependency inspection and rendering. A DragPoint captures the reactive identity directly and declares the mathematical view used to map pointer positions back to exact rational coordinates.

Runnable RiXRun cell

.Plugin.Load("geometry"); view2 := [-2,-2,6,4]; size2 := [640,480]; $$a2 := {: 0,0}; $$graph2 := .geometry.ConstructionGraph([ {= id=:a,free=1,value=.geometry.Point($a2[1],$a2[2]) }, {= id=:b,dependsOn=[:a],construct=(values)-> .geometry.Point(values[:a][:x]+3,values[:a][:y]+1) }, {= id=:ab,dependsOn=[:a,:b],construct=(values)-> .geometry.Line(values[:a],values[:b]) } ]); handle2 := .Graphics.DragPoint({= target=$$a2,label="Move a",coordinateSystem={= view=view2,size=size2 }, style={= fill="#7c3aed",stroke="#ffffff",width=2,hitId="a:handle" } }); $$workbench2 := .geometry.Workbench($graph2,{= view=view2,size=size2,handles=[{= id=:a,graphic=handle2 }] }); $workbench2;

RiX Web supplies the object tree, property inspector, keyboard traversal, session movement history, and export button. The same value still renders as ordinary Graphics in static hosts.

Add exact free points from the canvas

Positioned Graphics.Action nodes translate a pointer or keyboard cursor through the declared mathematical coordinate system before invoking their RiX callback. The point tool below appends a retained free node; its companion actions replay the same exact create history.

Runnable RiXRun cell

.Plugin.Load("geometry"); view3 := [-4,-3,4,3]; size3 := [640,480]; $$graph3 := .geometry.ConstructionGraph([]); actions3 := [ .Graphics.Action({= id="geometry-author-point",target=$$graph3, action=(current,position)->.geometry.AddPoint( current,.geometry.Point(position[1],position[2]),{= snap=1/4,maxNodes=32 } ), label="Add an exact free point",coordinateSystem={= view=view3,size=size3 }, children=[.Graphics.Rectangle([0,0],size3,{= fill="transparent",stroke="none" })] }), .Graphics.Action({= id="geometry-author-line",target=$$graph3, action=(current,ids)->.geometry.AddLine(current,ids[1],ids[2]),children=[] }), .Graphics.Action({= id="geometry-author-circle",target=$$graph3, action=(current,ids)->.geometry.AddCircle(current,ids[1],ids[2]),children=[] }), .Graphics.Action({= id="geometry-author-intersection",target=$$graph3, action=(current,ids)->.geometry.AddIntersection(current,ids[1],ids[2]),children=[] }), .Graphics.Action({= id="geometry-author-measurement",target=$$graph3, action=(current,ids)->.geometry.AddMeasurement(current,ids[1],ids[2]),children=[] }), .Graphics.Action({= id="geometry-author-undo",target=$$graph3, action=current->.geometry.Undo(current),children=[] }), .Graphics.Action({= id="geometry-author-redo",target=$$graph3, action=current->.geometry.Redo(current),children=[] }) ]; $$workbench3 := .geometry.AuthoringWorkbench($graph3,actions3,{= view=view3,size=size3,snap=1/4,maxNodes=32 }); $workbench3;

Click empty canvas space to place p1, p2, and so on. Select Point tool to focus its surface; arrows move the cursor, Shift-arrows move ten pixels, and Enter or Space places the point. Select Line tool or Circle tool, then choose two distinct points in the construction tree. For a circle, choose its center first and its through-point second. Intersection tool accepts two lines, circles, or conics; Distance tool accepts two points. snap is applied in mathematical coordinates, and maxNodes is a strict construction bound.

Keep unresolved intersections visible

Parallel lines return an intersection result whose status is parallel. Including that result in a drawing produces a visible diagnostic rather than a fabricated point.

Runnable RiXRun cell

.Plugin.Load("geometry"); first4 := .geometry.Line(.geometry.Point(0, 0), .geometry.Point(4, 0)); second4 := .geometry.Line(.geometry.Point(0, 2), .geometry.Point(4, 2)); unresolved4 := .geometry.Intersect(first4, second4); status4 := .geometry.Status(unresolved4); .geometry.Draw([first4, second4, unresolved4], {= view=[-1,-1,5,3], size=[600,400] });

Transform a polygon and check a constraint

Transforms preserve exact coordinates. Constraints retain their exact residual instead of turning a drawing into a Boolean-only result.

Runnable RiXRun cell

.Plugin.Load("geometry"); a5 := .geometry.Point(0,0); b5 := .geometry.Point(4,0); c5 := .geometry.Point(1,3); triangle5 := .geometry.Polygon([a5,b5,c5]); shiftAndScale5 := .geometry.Affine([[3/2,0,1],[0,3/2,2]]); moved5 := .geometry.Transform(triangle5,shiftAndScale5); base5 := .geometry.Line(a5,b5); onBase5 := .geometry.Constraint(:onLine,[.geometry.Point(2,0),base5]); .geometry.Draw([triangle5,moved5],{= view=[-1,-1,8,8],size=[560,560] });

Measure and rotate without binary floats

Named measurements preserve the geometry kernel's exact/certified split. Arbitrary rotations accept radians, degrees, or fractions of a turn.

Runnable RiXRun cell

.Plugin.Load("geometry"); origin6 := .geometry.Point(0,0); p6 := .geometry.Point(4,0); segment6 := .geometry.Segment(origin6,p6); length6 := .numerics.Refine(.geometry.Length(segment6),{= absoluteWidth=1/1000,maxWork=32 }); rotation6 := .geometry.Rotate(origin6,1/6,:turns); image6 := .geometry.Transform(p6,rotation6); .Fragment([ .Table(["quantity","value"],[["squared length",.geometry.SquaredDistance(origin6,p6)],["length candidate",length6[:approximation].Candidate()]]), .geometry.Draw([p6,image6],{= view=[-1,-1,5,5],size=[360,360] }) ]);

Quarter turns are rational affine maps. Rational fractions of a turn carry exact algebraic coordinates; arbitrary radians carry certified-real coordinates and can be refined or drawn.

Explore the algebraic lane

A compass construction may leave rational coordinates without leaving exact mathematics. The diagonal through a rational circle meets it at coordinates involving sqrt(2). Exact polynomial certificates decide the intersection topology and retain exact algebraic points.

Runnable RiXRun cell

.Plugin.Load("geometry"); circle7 := .geometry.Circle(.geometry.Point(0,0),2); diagonal7 := .geometry.Line(.geometry.Point(-2,-2),.geometry.Point(2,2)); meeting7 := .geometry.Intersect(diagonal7,circle7); eighthTurn7 := .geometry.CircularAngle(1/8,:turns); {= intersectionStatus=meeting7[:status], coordinateDomain=meeting7[:points][1][:coordinateDomain], angleTurns=eighthTurn7[:turns], angleDomain=eighthTurn7[:coordinateDomain] };

A rational turn is exact circular data. Its sine and cosine are algebraic because they are coordinates of a root of unity. The nonzero radian magnitude 2*pi*p/q is transcendental, so the plugin does not mislabel it as an algebraic real. A rational number supplied directly in radians is generally not a rational turn and stays on the general certified lane.

Algebraic coefficients make intersection equality, sign, and multiplicity decidable. With arbitrary certified-real coefficients, interval refinement can prove separation from zero but cannot always prove exact zero. Those intersections explicitly return undecided with evidence when their budget is exhausted.

Inspect certified intersection decisions

Line-conic intersections retain both the exact discriminant sign and the Sturm root-count certificate used to classify the result.

Runnable RiXRun cell

.Plugin.Load("geometry"); circle8 := .geometry.Circle(.geometry.Point(0,0),1); line8 := .geometry.Line(.geometry.Point(-2,2),.geometry.Point(2,2)); intersection8 := .geometry.Intersect(line8,circle8); {= status=intersection8[:status], discriminantSign=intersection8[:evidence][:discriminantSign], rootCount=intersection8[:evidence][:rootCount] };

Refine an implicit conic with bounded work

The implicit value stays mathematical and serializable. Refine creates one portable snapshot for the requested viewport while retaining work and uncertainty records.

Runnable RiXRun cell

.Plugin.Load("geometry"); ellipse9 := .geometry.Implicit({= coefficients=[1,0,4,0,0,-4], domain=[-3,-2,3,2], style={= stroke="#0f766e",width=2 } }); result9 := .geometry.Refine(ellipse9,{= viewport={= x=[-3,3],y=[-2,2] }, size=[600,400], tolerance=1/8, maxWork=1200 }); .Fragment([ .Figure(result9[:graphic],"Bounded implicit ellipse"), .Table({= columns=["resolved","uncertain cells","work"],rows=[[ result9[:resolved],result9[:uncertainty].Len(),result9[:work][:cells] ]] }) ]);