AccelStepper: smooth stepper moves that ramp up and down

// Stepper turntable: a photo turntable on a 28BYJ-48 and its ULN2003 board,
// with AccelStepper. Press the button and it turns the plate a full circle in
// eight stops, pausing at each one while the LED fires the camera.
//
//   ULN2003  IN1 pin 8, IN2 pin 9, IN3 pin 10, IN4 pin 11, + to 5V, - to GND
//   button   pin 2 to GND, INPUT_PULLUP
//   LED      pin 13, the camera trigger (the Uno's own LED)
//
// The pins go to AccelStepper as 8, 10, 9, 11: IN1, IN3, IN2, IN4. The library
// numbers its four pins for a motor on H bridges, and that order is what makes
// its sequence walk a ULN2003's four coils round in order. Given 8, 9, 10, 11
// the motor hums and stays where it is.
//
// HALF4WIRE is half stepping: 4096 steps to a turn of the output shaft, so a
// stop every eighth of a turn is 512 steps. The plate speeds up and slows down
// for each stop, which is what keeps a real one from sliding the object on it.
// run() makes at most one step a call, so the loop never waits while the
// plate is moving.

#include <AccelStepper.h>

const int BUTTON = 2;
const int SHUTTER = 13;
const int STOPS = 8;
const long STEPS_PER_STOP = 4096 / STOPS;  // 4096 half steps to a turn

AccelStepper plate(AccelStepper::HALF4WIRE, 8, 10, 9, 11);

int stop = 0;
bool turning = false;

void setup() {
  pinMode(BUTTON, INPUT_PULLUP);
  pinMode(SHUTTER, OUTPUT);
  plate.setMaxSpeed(800);
  plate.setAcceleration(400);
  Serial.begin(115200);
  Serial.println("turntable: press the button to start a circle");
}

void photo() {
  // One digit is all eight stops need.
  Serial.print("photo ");
  Serial.write('1' + stop);
  Serial.println();
  digitalWrite(SHUTTER, HIGH);
  delay(200);
  digitalWrite(SHUTTER, LOW);
  delay(300);
}

void loop() {
  if (!turning) {
    if (digitalRead(BUTTON) == LOW) {
      turning = true;
      stop = 0;
      plate.setCurrentPosition(0);
      photo();
      plate.moveTo(STEPS_PER_STOP);
    }
    return;
  }
  if (plate.distanceToGo() != 0) {
    plate.run();
    return;
  }
  stop++;
  if (stop < STOPS) {
    photo();
    plate.moveTo((stop + 1) * STEPS_PER_STOP);
  } else {
    Serial.println("full circle: done");
    plate.disableOutputs();
    turning = false;
  }
}