Creating a concrete curb with curves is a simple task if you use a curb machine rather than wooden forms for curb setup. A curb machine is a concrete curb extruder, pumping out the curb in a set shape along any path you choose. If the path is curved, you simply follow the curved path, creating a curved curb in the process. After extrusion, there's a short finishing process and then when the curb cures you'll have the curb shape you wish, curves and all.
Apply epoxy sealant to the concrete where you're placing your curved curb as a binding agent between the new concrete of the curb and the old concrete. Use a 1-inch-wide brush and place the epoxy in two parallel strips running the full length of your proposed curb, each strip positioned so that it's about one inch from the outside edges of the curb.
Mix the concrete for your curb in a wheelbarrow using a hoe, adding just enough water to the concrete mix to create a 0-slump thickness. Test the slump of the concrete by placing the mix in a Styrofoam cup that you've cut the bottom from, turning the cup upside down on a flat surface and removing the cup. A 0-slump mix will stand stiffly without losing its form at all after the removal of the cup.
Place the curb mold attachment onto the curbing machine. This mold will create the shape of your curb. Check the machine manufacturer's instructions for the attachment procedure to ensure that the attachment is set properly for use.
Push the curbing machine to the point where you wish to begin the curb. Check that the epoxy is at the tacky stage, not wet but still sticky, and wait until it reaches the stage to begin laying the curb. It should take no longer than 10 minutes total from application to tackiness.
Fill the concrete reservoir of the curbing machine with concrete from the wheelbarrow using a spade. Start the curbing machine and begin extruding the curb from the mold attachment. Move the curbing machine along the epoxy strip, following the strip to lay your curb. The machine is on wheels, so all that's required for the curve is to turn the machine along the curving line as you follow the epoxy, extruding the curb behind the machine.
Continue to fill the concrete reservoir with concrete to continue forming the curb. Stop filling as you begin to approach the end of the curb, and then allow the reservoir to empty as you reach the end in order to stop the curb. Remove the machine from the curb line after completion for full access to the curb end for finishing.
Smooth the face of the curb along its length by dragging a front trowel with the same shape as the mold attachment front along the curb. This will remove any excess material and give the curb a consistent face. Position a firm piece of plastic at the rear of the curb, running along the rear as you smooth the face to help support the concrete during the smoothing process. Repeat the smoothing process with the rear of the curb switching the front trowel for the back trowel instead.
Complete the curb by cutting expansion joints into the material, spaced three feet apart along its length. Use a putty knife to cut the joints at least two inches deep into the curb to prevent the curb from cracking with seasonal changes in temperature.