Rectangle-to-Circular Transition: flat pattern
Square-to-round transitions with flat sides and conical corners, offset or inclined. Square-to-round transition: flat triangular sides and conical corners, developed by triangulation.
Open the rectangle-to-circular transition in SheetMorphConfigurations
- Concentric. Rectangle and circle share one centreline. End planes are parallel.
- Centre offset. The circle centre is displaced from the rectangle centre by dx, dy. End planes are parallel.
- Eccentric. The circle is tangent to the plane of the selected rectangle side(s); the centre offset is calculated.
- Angled faces. One or both end planes are inclined. End centres stay on the Z axis.
- Combined offset + angle. Inclined end planes together with centre offsets or flush sides.
A worked example
The concentric configuration with the sizes SheetMorph opens with, in mm. Triangulated approximation: each conical corner is 8 flat facets: 8 parts, 1 seam.
| Blank, the rectangle the plate is cut from | 1217.3 mm × 506.3 mm |
| Length of cut, every outline and hole | 2893.1 mm |
| Bends (4) down, the pattern drawn outside face up | 6.4° |
| Bends (4) down, the pattern drawn outside face up | 5.9° |
| End A width (W) | 400.00 mm |
| End A height (H) | 300.00 mm |
| End B diameter (D) | 250.00 mm |
| End A width (W) | 401.00 mm |
| End A height (H) | 301.00 mm |
| End B diameter (D) | 251.00 mm |
| Length along Z (L) | 350.00 mm |
| Material thickness (T) | 1.00 mm |
| Horizontal centre offset (dx) | 0.00 mm |
| Vertical centre offset (dy) | 0.00 mm |
| Centreline length | 350.00 mm |
| Centreline angle to Z | 0.00 ° |
| Angle between end planes | 0.00 ° |
| Divisions (total) | 32 |
| Chord between divisions | 24.60 mm |
| Radius of division points | 125.50 mm |
| True circumference | 788.54 mm |
| Developed round edge | 787.27 mm |
| Developed minus true | -1.27 mm |
| Chord-to-arc gap | 0.60 mm |
| Bottom midline M0 - B8 | 350.89 mm |
| Right midline M1 - B16 | 357.95 mm |
| Top midline M2 - B24 | 350.89 mm |
| Left midline M3 - B0 | 357.95 mm |
| Bottom panel area | 70,354 mm² |
| Right panel area | 53,871 mm² |
| Top panel area | 70,354 mm² |
| Left panel area | 53,871 mm² |
| A0 - B0 | 388.30 mm |
| A0 - B1 | 379.96 mm |
| A0 - B2 | 374.37 mm |
| A0 - B3 | 371.85 mm |
| A0 - B4 | 372.56 mm |
| A0 - B5 | 376.47 mm |
| A0 - B6 | 383.33 mm |
| A0 - B7 | 392.73 mm |
| A0 - B8 | 404.14 mm |
| A1 - B8 | 404.14 mm |
| A1 - B9 | 392.73 mm |
| A1 - B10 | 383.33 mm |
| A1 - B11 | 376.47 mm |
| A1 - B12 | 372.56 mm |
| A1 - B13 | 371.85 mm |
| A1 - B14 | 374.37 mm |
| A1 - B15 | 379.96 mm |
| A1 - B16 | 388.30 mm |
| A2 - B16 | 388.30 mm |
| A2 - B17 | 379.96 mm |
| A2 - B18 | 374.37 mm |
| A2 - B19 | 371.85 mm |
| A2 - B20 | 372.56 mm |
| A2 - B21 | 376.47 mm |
| A2 - B22 | 383.33 mm |
| A2 - B23 | 392.73 mm |
| A2 - B24 | 404.14 mm |
| A3 - B24 | 404.14 mm |
| A3 - B25 | 392.73 mm |
| A3 - B26 | 383.33 mm |
| A3 - B27 | 376.47 mm |
| A3 - B28 | 372.56 mm |
| A3 - B29 | 371.85 mm |
| A3 - B30 | 374.37 mm |
| A3 - B31 | 379.96 mm |
| A3 - B0 | 388.30 mm |
| Panels (flat + conical) | 8 |
| Triangular facets | 40 |
| Pattern extent X | 1217.28 mm |
| Pattern extent Y | 506.27 mm |
| Developed area | 391,958 mm² |
| Total cut length | 2893.06 mm |
| Total seam (joint) length | 350.89 mm |
How it is developed
Triangulation (true-length unfolding). This is a triangulated approximation: the curved surface is divided into small flat triangles, each laid out at its true size. The finer the division, the closer the pattern is to the surface; SheetMorph states the deviation it found.
- The four flat triangular panels are planar and develop exactly.
- The four corners are parts of oblique cones. Each is approximated by 8 narrow triangles from the rectangle corner to the circle divisions; the pattern is exact for that faceted surface.
- Division points lie on the true circle, so the developed round edge (a chain of chords) is slightly shorter than the true circumference. Increase the divisions, or use the matched-circumference model, if the joint to the round duct must close exactly.
- 8 bend lines separate flats from corners; 28 forming lines inside the corners each take a small bend.
- One-piece wrap-around pattern with 1 seam along the middle of a flat panel.
Checked before it can be exported
Every pattern SheetMorph draws is checked against the 3D part it comes from. For this example:
- End A is a true rectangle (passed): Side lengths and diagonals match the specified width and height, and each side midpoint bisects its side (max deviation 0).
- End B points on one circle, equally spaced (passed): 32 division points lie in the End B plane at radius 125.5 with equal chords (max deviation 1.17e-13). Points are on the true circle; the developed round edge is a chain of chords.
- Surface connectivity (passed): 40 facets form one closed band: every edge inside a part joins exactly two facets, every open edge one (V - E + F = 0).
- Developed edges equal true 3D lengths (passed): 120 facet edges compared between the 3D surface and the flat pattern (max deviation 2.27e-13).
- Shared edges and seam closure (passed): 39 internal edges are shared vertex-for-vertex between adjacent facets; 1 seam(s) have mating edges of equal true length (max deviation 1.71e-13).
- Developed facets consistently oriented (passed): All developed facets are counter-clockwise (outside face up); none is mirrored or degenerate.
- Developed area equals surface area (passed): Facet area in 3D and in the flat pattern agree (difference 5.82e-11).
- Cut outlines closed and non-overlapping (passed): 1 closed outline(s); no outline crosses itself or another outline.
- Expected symmetry (passed): left/right and top/bottom mirror symmetry holds: every mirrored corner and circle point coincides (max deviation 1.81e-13).
Assumptions
- The pattern is developed on the mid-thickness surface. Inside or outside end dimensions are converted to it by one thickness; mid-thickness dimensions are used unchanged.
- Bends are sharp folds of the mid-thickness surface (equivalent to a neutral axis at mid-thickness); no bend radius or separate bend allowance is applied.
- No seam, lap, flange, weld or trimming allowance is added to any outline.
- The circle is represented by equally spaced division points joined by straight chords.
- Each rectangle corner fans to one quarter of the circle; each rectangle side joins the quadrant point facing it.
- End-plane twist about the Z axis is not supported.
- The pattern is drawn outside face up.
Limits of this module
- No bend radius or separate k-factor: bend and forming lines are sharp folds of the mid-thickness surface.
- Inside/outside conversion is one thickness at each end; the extra difference on steeply inclined walls is not modelled.
- No seam, lap, flange, weld or trim allowances; no straight collars on either end.
- The round edge of the pattern is a chain of chords, not a smooth curve.
- No end-plane twist about the Z axis.
- Sheet nesting, grain direction and stock-size checks are not performed.
Conventions
Coordinate system. Right-handed. X is horizontal (width, +X right), Y is vertical (height, +Y top), Z is the reference axis. End A, the rectangle, is centred on the origin; End B, the circle, is centred on (dx, dy, L). Left, right, top and bottom are as seen looking from End B back toward End A.
Length and offsets. L is the distance between the two end centres measured along Z, not the slant distance. dx and dy are the circle centre minus the rectangle centre: positive dx moves the circle right, positive dy moves it up.
End-plane inclination. Each end can be inclined about two axes through its own centre, so the centre does not move. "About X" turns the end about the horizontal axis; positive moves its top toward +Z. "About Y" turns it about the vertical axis; positive moves its right side toward -Z. The rotation about X is applied first. Limit: +/-75 deg. The inclined end stays a true rectangle or circle.
Flush sides. Making a side flush (left, right, top or bottom) places the circle tangent to the plane of that rectangle side, so that side of the transition is straight. The centre offset on that axis is then calculated: without inclination, left gives dx = D/2 - W/2 and bottom gives dy = D/2 - H/2 (right and top the opposite sign). Left/right is rejected if the rectangle is inclined about both axes.
Numbering. Rectangle corners: A0 bottom-left, A1 bottom-right, A2 top-right, A3 top-left. Side midpoints: M0 bottom, M1 right, M2 top, M3 left. Circle divisions B0, B1, ... run counter-clockwise (seen from End B) starting at the left quadrant point; with N divisions per quarter the quadrant points are B0 left, BN bottom, B2N right, B3N top.
Panels. Flat panels P0 Bottom, P1 Right, P2 Top, P3 Left each join a rectangle side to the quadrant point facing it. Conical corner panels C0_BL, C1_BR, C2_TR, C3_TL each join a rectangle corner to a quarter of the circle.
Development method. Each conical corner is divided into N narrow triangles from the rectangle corner to consecutive circle divisions. All triangles are unfolded in sequence from their true 3D edge lengths, each sharing an edge with the previous one. Flat panels are exact; the corners are a faceted approximation of a cone, and the round edge of the pattern is a chain of straight chords.
Circle model. Inscribed: division points are on the true circle, and the developed round edge is slightly shorter than pi x D (0.16% with 8 divisions per quarter). Matched circumference: points are on a marginally larger circle so the developed round edge equals pi x D exactly. The difference is reported in the calculation panel.
Seam and layout. The seam runs along the middle of a flat panel, from the midpoint of a rectangle side to the quadrant point facing it. One piece has one seam; two halves have seams on two opposite sides; four quarters, for work too big for one sheet, have a seam on every side.
Material thickness. A flat pattern has the length of the sheet's neutral surface, taken as mid-thickness. Say whether your end dimensions are inside, mid-thickness or outside sizes: inside sizes are increased by one thickness and outside sizes reduced by one thickness to reach the mid-thickness surface, on which the pattern is developed. Length, offsets and inclinations are not affected. Sharp folds of the mid-thickness surface correspond to a neutral axis at mid-thickness; bend radii are not modelled.
Change the sizes to your own in SheetMorph, check the drawing, add seam and flange allowances, and export DXF, PDF, SVG, CSV or G-code. Check every pattern against your own standards before cutting.
Open the rectangle-to-circular transitionOther transitions
- Rectangle to Rectangle: Hoppers and duct transitions between two rectangular ends, centred, offset or inclined.
- Round to Oval: A round end to a flat-oval end: flat triangles under the straight sides, curved parts round the ends.
- Rectangle to Oval: A rectangular end to a flat-oval end: flat panels on the long sides and parts of cones at the corners.
- Rectangle to Ellipse: A rectangular end to an elliptical end, or an ellipse to a rectangle: concentric, offset, eccentric or with angled faces.
- Polygon: A regular polygon, such as a hexagon or an octagon, to a circle or to another polygon.
- Twisted Rectangle: A transition between two rectangles, one of them turned about the axis; its twisted sides are split into flat triangles.
- Pyramid Hopper: A four-sided pyramid between two rectangles, such as a hopper or a plenum, with or without flanges turned outward.