• SPI_WRITE16() optimization for Adafruit GFX
• moononournation/Arduino_GFX: Rewrite of Adafruit GFX for speed
• GFX-Forever: Another optimized library

1. X(𝑡) = cos(𝑡) − 0.5cos(5𝑡)
2. Y(𝑡) = sin(𝑡) + 0.5sin(5𝑡)
3. Z(𝑡) = 1.32204cos(3𝑡)

Where 𝑡 is in degrees, i.e., 𝑡 ∈ [0,360)

To normalize the diameter to '1' Use the following equation

1. X(𝑡) = 0.378122(cos(t) − 0.5cos(5t))
2. Y(𝑡) = 0.378122(sin(t) + 0.5sin(5t))
3. Z(𝑡) = 0.5cos(3t)

​

Xt = 0.378122*(cos(t)−0.5*cos(5*t));
Yt = 0.378122*(sin(t)+0.5*sin(5*t));
Zt = 0.5*cos(3t);

Arduino_GFX doesn't let you draw lines with thickness, here I have two functions, one thats “simple” and one thats a bit more optimized. I like to chain lines together to make a polyline so rounded edges make it look better.

This is a “naive” function which computes two triangles representing the thickened line and then plops two circles on the ends. It gets the job done, but it isn't efficient.

// Draw a thick line with rounded ends using Adafruit_GFX primitives
void DrawNaiveThickLineRounded(int x1, int y1, int x2, int y2, int thickness, int color) {
  // Calculate the differences between endpoints
  int deltaX = x1 - x2;
  int deltaY = y1 - y2;
 
  // Compute the length once
  float len = sqrt(sq(deltaX) + sq(deltaY));
 
  // If the endpoints are nearly identical, draw a single circle
  if (len < 0.001) {
    tft.fillCircle(x1, y1, thickness / 2, color);
    return;
  }
 
  // Compute factor for the perpendicular offset
  float factor = (thickness / 2.0) / len;
 
  // Calculate the perpendicular offsets for rounded thickness
  float offsetX = factor * (y1 - y2);
  float offsetY = factor * (x1 - x2);
 
  // Draw the thick line as two triangles forming a rectangle
  tft.fillTriangle(x1 + offsetX, y1 - offsetY,
                   x1 - offsetX, y1 + offsetY,
                   x2 + offsetX, y2 - offsetY, color);
  tft.fillTriangle(x1 - offsetX, y1 + offsetY,
                   x2 - offsetX, y2 + offsetY,
                   x2 + offsetX, y2 - offsetY, color);
 
  // Add rounded caps at both endpoints
  tft.fillCircle(x1, y1, thickness / 2, color);
  tft.fillCircle(x2, y2, thickness / 2, color);
}

This is more optimized code focusing on eliminating redundant calculations.

// Draw a thick line with rounded ends using a unified scanline fill approach.
// Focuses more on integer optimization, further optimization can be made by eliminating division.
void DrawThickLineRoundedHLines(int x1, int y1, int x2, int y2, int thickness, uint16_t color) {
  // Half thickness (using integer division is fine for even thicknesses)
  int r = thickness >> 1;
  int dxLine = x2 - x1;
  int dyLine = y2 - y1;
  int d2 = dxLine * dxLine + dyLine * dyLine;
 
  // If endpoints coincide, draw a full circle.
  if (d2 == 0) {
    tft.fillCircle(x1, y1, r, color);
    return;
  }
 
  // Compute the length and normalized line direction.
  float len = sqrt((float)d2);
  float lx = dxLine / (float)len;
  float ly = dyLine / (float)len;
 
  // Compute perpendicular offset (using (dy, -dx)) scaled to half thickness.
  float factor = (thickness / 2.0f) / len;
  float offX = factor * dyLine;
  float offY = factor * -dxLine;
 
  // Compute the four vertices (of the quadr) in clockwise order.
  // These define the main body of the line.
  float Ax = x1 + offX, Ay = y1 + offY;
  float Bx = x1 - offX, By = y1 - offY;
  float Cx = x2 - offX, Cy = y2 - offY;
  float Dx = x2 + offX, Dy = y2 + offY;
 
  // Determine overall bounding box (including the circular caps).
  float bboxMinX = min(min(min(Ax, Bx), min(Cx, Dx)), (float)min(x1 - r, x2 - r));
  float bboxMaxX = max(max(max(Ax, Bx), max(Cx, Dx)), (float)max(x1 + r, x2 + r));
  float bboxMinY = min(min(min(Ay, By), min(Cy, Dy)), (float)min(y1 - r, y2 - r));
  float bboxMaxY = max(max(max(Ay, By), max(Cy, Dy)), (float)max(y1 + r, y2 + r));
 
  int iMinY = (int)floor(bboxMinY);
  int iMaxY = (int)ceil(bboxMaxY);
 
  // Lambda to process an edge of the quadr.
  // Returns the x coordinate where the horizontal scanline at 'y' intersects the edge,
  // or NAN if there is no intersection.
  auto processEdge = [=](float x0, float y0, float x1, float y1, int y) -> float {
    if ((y0 <= y && y1 > y) || (y1 <= y && y0 > y)) {
      float t = (y - y0) / (y1 - y0);
      return x0 + t * (x1 - x0);
    }
    return NAN;
  };
 
  // For each scanline, we compute up to three segments: one from the quadr body,
  // one from the start cap, and one from the end cap.
  // We then merge overlapping segments and draw them.
  struct Segment { float x0, x1; };
 
  for (int y = iMinY; y <= iMaxY; y++) {
    Segment segments[3];
    int segCount = 0;
 
    // 1. Quadr segment: process all four edges of the quadr.
    float inters[4];
    int count = 0;
    float ttemp;
    ttemp = processEdge(Ax, Ay, Bx, By, y); if (!isnan(ttemp)) inters[count++] = ttemp;
    ttemp = processEdge(Bx, By, Cx, Cy, y); if (!isnan(ttemp)) inters[count++] = ttemp;
    ttemp = processEdge(Cx, Cy, Dx, Dy, y); if (!isnan(ttemp)) inters[count++] = ttemp;
    ttemp = processEdge(Dx, Dy, Ax, Ay, y); if (!isnan(ttemp)) inters[count++] = ttemp;
    if (count >= 2) {
      float qx0 = inters[0], qx1 = inters[0];
      for (int i = 1; i < count; i++) {
        if (inters[i] < qx0) qx0 = inters[i];
        if (inters[i] > qx1) qx1 = inters[i];
      }
      segments[segCount++] = { qx0, qx1 };
    }
 
    // 2. Start cap (centered at (x1,y1)): if y is within the circle's vertical span.
    if (y >= y1 - r && y <= y1 + r) {
      float dyCap = y - y1;
      float dxCap = sqrt((float)(r*r - dyCap*dyCap));
      float capStart = x1 - dxCap;
      float capEnd   = x1 + dxCap;
      // Clip to the half circle: include only points for which dot((x-x1,y-y1),(lx,ly)) < 0.
      // Solve: (x - x1)*lx + (y - y1)*ly < 0  => if lx != 0, x < x1 - ((y-y1)*ly)/lx when lx > 0,
      // or x > x1 - ((y-y1)*ly)/lx when lx < 0.
      if (fabs(lx) > 1e-6) {
        float xBoundary = x1 - ((y - y1) * ly) / lx;
        if (lx > 0) {
          capEnd = min(capEnd, xBoundary);
        } else {
          capStart = max(capStart, xBoundary);
        }
      } else {
        // For near-vertical lines, if the condition isn't met, skip drawing the cap on this scanline.
        if ((ly > 0 && y >= y1) || (ly < 0 && y <= y1)) {
          capStart = 1, capEnd = 0; // empty
        }
      }
      if (capEnd >= capStart)
        segments[segCount++] = { capStart, capEnd };
    }
 
    // 3. End cap (centered at (x2,y2)): if y is within the circle's vertical span.
    if (y >= y2 - r && y <= y2 + r) {
      float dyCap = y - y2;
      float dxCap = sqrt((float)(r*r - dyCap*dyCap));
      float capStart = x2 - dxCap;
      float capEnd   = x2 + dxCap;
      // Clip to the half circle: include only points for which dot((x-x2,y-y2),(lx,ly)) > 0.
      if (fabs(lx) > 1e-6) {
        float xBoundary = x2 - ((y - y2) * ly) / lx;
        if (lx > 0) {
          capStart = max(capStart, xBoundary);
        } else {
          capEnd = min(capEnd, xBoundary);
        }
      } else {
        if ((ly > 0 && y <= y2) || (ly < 0 && y >= y2)) {
          capStart = 1, capEnd = 0;
        }
      }
      if (capEnd >= capStart)
        segments[segCount++] = { capStart, capEnd };
    }
 
    // If no segments were found on this scanline, continue.
    if (segCount == 0)
      continue;
 
    // Merge segments: sort by starting x and then combine overlapping ones.
    // (Since there are few segments, a simple bubble sort is sufficient.)
    for (int i = 0; i < segCount - 1; i++) {
      for (int j = i + 1; j < segCount; j++) {
        if (segments[j].x0 < segments[i].x0) {
          Segment tmp = segments[i];
          segments[i] = segments[j];
          segments[j] = tmp;
        }
      }
    }
 
    // Merge overlapping segments.
    float mergedStart = segments[0].x0;
    float mergedEnd = segments[0].x1;
    for (int i = 1; i < segCount; i++) {
      if (segments[i].x0 <= mergedEnd + 1) {  // overlapping or contiguous
        if (segments[i].x1 > mergedEnd)
          mergedEnd = segments[i].x1;
      } else {
        // Draw the current merged segment.
        int ix0 = (int)ceil(mergedStart);
        int ix1 = (int)floor(mergedEnd);
        if (ix1 >= ix0)
          tft.drawFastHLine(ix0, y, ix1 - ix0 + 1, color);
        mergedStart = segments[i].x0;
        mergedEnd = segments[i].x1;
      }
    }
    // Draw the final merged segment.
    int ix0 = (int)ceil(mergedStart);
    int ix1 = (int)floor(mergedEnd);
    if (ix1 >= ix0)
      tft.drawFastHLine(ix0, y, ix1 - ix0 + 1, color);
  }
}

In practice

#include <Adafruit_GFX.h>
#include <Adafruit_ST7796S.h>

#define TFT_CS         2
#define TFT_RST        -1
#define TFT_DC         4
Adafruit_ST7796S tft = Adafruit_ST7796S(TFT_CS, TFT_DC, TFT_RST);

#define SCREEN_WIDTH  480
#define SCREEN_HEIGHT 360

GFXcanvas16 canvas(SCREEN_WIDTH, SCREEN_HEIGHT);  // Off-screen framebuffer

uint8_t rotate = 0;

const float PI_F     = 3.14159265;
const float SCALE    = 0.378122;
const float Z_SCALE  = 0.5;
const float Z_OFFSET = 2.0;
const int   NUM_SEGMENTS = 72;

const uint16_t blackish = 0x0000;

float rotation = 0.0;

struct Segment3D {
  int x1, y1, x2, y2;
  float avgZ;
  int thickness;
};
Segment3D segments[NUM_SEGMENTS];

struct Vec3 {
  float x, y, z;
};

Vec3 rotateY(const Vec3& v, float angleDeg) {
  float a = angleDeg * PI_F / 180.0;
  float cosA = cos(a);
  float sinA = sin(a);
  return {
    cosA * v.x + sinA * v.z,
    v.y,
    -sinA * v.x + cosA * v.z
  };
}

void project(const Vec3& pt, int &px, int &py) {
  float f = 1.0 / (Z_OFFSET - pt.z);
  float x_proj = pt.x * f;
  float y_proj = pt.y * f;
  px = (int)((x_proj + 1.0) * SCREEN_WIDTH / 2.0);
  py = (int)((1.0 - y_proj) * SCREEN_HEIGHT / 2.0);
}

inline uint16_t rgb888_to_565(uint8_t r, uint8_t g, uint8_t b) {
  uint16_t r5 = (r * 249 + 1014) >> 11;
  uint16_t g6 = (g * 253 + 505) >> 10;
  uint16_t b5 = (b * 249 + 1014) >> 11;
  return (r5 << 11) | (g6 << 5) | b5;
}

void DrawThickLineRoundedHLines(GFXcanvas16& canvas, int x1, int y1, int x2, int y2, int thickness, uint16_t color) {// Draw a thick line with rounded ends using a unified scanline fill approach.
  // Half thickness (using integer division is fine for even thicknesses)
  int r = thickness >> 1;
  int dxLine = x2 - x1;
  int dyLine = y2 - y1;
  int d2 = dxLine * dxLine + dyLine * dyLine;
 
  // If endpoints coincide, draw a full circle.
  if (d2 == 0) {
    canvas.fillCircle(x1, y1, r, color);
    return;
  }
 
  // Compute the length and normalized line direction.
  float len = sqrt((float)d2);
  float lx = dxLine / (float)len;
  float ly = dyLine / (float)len;
 
  // Compute perpendicular offset (using (dy, -dx)) scaled to half thickness.
  float factor = (thickness / 2.0f) / len;
  float offX = factor * dyLine;
  float offY = factor * -dxLine;
 
  // Compute the four vertices (of the quadr) in clockwise order.
  // These define the main body of the line.
  float Ax = x1 + offX, Ay = y1 + offY;
  float Bx = x1 - offX, By = y1 - offY;
  float Cx = x2 - offX, Cy = y2 - offY;
  float Dx = x2 + offX, Dy = y2 + offY;
 
  // Determine overall bounding box (including the circular caps).
  float bboxMinX = min(min(min(Ax, Bx), min(Cx, Dx)), (float)min(x1 - r, x2 - r));
  float bboxMaxX = max(max(max(Ax, Bx), max(Cx, Dx)), (float)max(x1 + r, x2 + r));
  float bboxMinY = min(min(min(Ay, By), min(Cy, Dy)), (float)min(y1 - r, y2 - r));
  float bboxMaxY = max(max(max(Ay, By), max(Cy, Dy)), (float)max(y1 + r, y2 + r));
 
  int iMinY = (int)floor(bboxMinY);
  int iMaxY = (int)ceil(bboxMaxY);
 
  // Lambda to process an edge of the quadr.
  // Returns the x coordinate where the horizontal scanline at 'y' intersects the edge,
  // or NAN if there is no intersection.
  auto processEdge = [=](float x0, float y0, float x1, float y1, int y) -> float {
    if ((y0 <= y && y1 > y) || (y1 <= y && y0 > y)) {
      float t = (y - y0) / (y1 - y0);
      return x0 + t * (x1 - x0);
    }
    return NAN;
  };
 
  // For each scanline, we compute up to three segments: one from the quadr body,
  // one from the start cap, and one from the end cap.
  // We then merge overlapping segments and draw them.
  struct Segment { float x0, x1; };
 
  for (int y = iMinY; y <= iMaxY; y++) {
    Segment segments[3];
    int segCount = 0;
 
    // 1. Quadr segment: process all four edges of the quadr.
    float inters[4];
    int count = 0;
    float ttemp;
    ttemp = processEdge(Ax, Ay, Bx, By, y); if (!isnan(ttemp)) inters[count++] = ttemp;
    ttemp = processEdge(Bx, By, Cx, Cy, y); if (!isnan(ttemp)) inters[count++] = ttemp;
    ttemp = processEdge(Cx, Cy, Dx, Dy, y); if (!isnan(ttemp)) inters[count++] = ttemp;
    ttemp = processEdge(Dx, Dy, Ax, Ay, y); if (!isnan(ttemp)) inters[count++] = ttemp;
    if (count >= 2) {
      float qx0 = inters[0], qx1 = inters[0];
      for (int i = 1; i < count; i++) {
        if (inters[i] < qx0) qx0 = inters[i];
        if (inters[i] > qx1) qx1 = inters[i];
      }
      segments[segCount++] = { qx0, qx1 };
    }
 
    // 2. Start cap (centered at (x1,y1)): if y is within the circle's vertical span.
    if (y >= y1 - r && y <= y1 + r) {
      float dyCap = y - y1;
      float dxCap = sqrt((float)(r*r - dyCap*dyCap));
      float capStart = x1 - dxCap;
      float capEnd   = x1 + dxCap;
      // Clip to the half circle: include only points for which dot((x-x1,y-y1),(lx,ly)) < 0.
      // Solve: (x - x1)*lx + (y - y1)*ly < 0  => if lx != 0, x < x1 - ((y-y1)*ly)/lx when lx > 0,
      // or x > x1 - ((y-y1)*ly)/lx when lx < 0.
      if (fabs(lx) > 1e-6) {
        float xBoundary = x1 - ((y - y1) * ly) / lx;
        if (lx > 0) {
          capEnd = min(capEnd, xBoundary);
        } else {
          capStart = max(capStart, xBoundary);
        }
      } else {
        // For near-vertical lines, if the condition isn't met, skip drawing the cap on this scanline.
        if ((ly > 0 && y >= y1) || (ly < 0 && y <= y1)) {
          capStart = 1, capEnd = 0; // empty
        }
      }
      if (capEnd >= capStart)
        segments[segCount++] = { capStart, capEnd };
    }
 
    // 3. End cap (centered at (x2,y2)): if y is within the circle's vertical span.
    if (y >= y2 - r && y <= y2 + r) {
      float dyCap = y - y2;
      float dxCap = sqrt((float)(r*r - dyCap*dyCap));
      float capStart = x2 - dxCap;
      float capEnd   = x2 + dxCap;
      // Clip to the half circle: include only points for which dot((x-x2,y-y2),(lx,ly)) > 0.
      if (fabs(lx) > 1e-6) {
        float xBoundary = x2 - ((y - y2) * ly) / lx;
        if (lx > 0) {
          capStart = max(capStart, xBoundary);
        } else {
          capEnd = min(capEnd, xBoundary);
        }
      } else {
        if ((ly > 0 && y <= y2) || (ly < 0 && y >= y2)) {
          capStart = 1, capEnd = 0;
        }
      }
      if (capEnd >= capStart)
        segments[segCount++] = { capStart, capEnd };
    }
 
    // If no segments were found on this scanline, continue.
    if (segCount == 0)
      continue;
 
    // Merge segments: sort by starting x and then combine overlapping ones.
    // (Since there are few segments, a simple bubble sort is sufficient.)
    for (int i = 0; i < segCount - 1; i++) {
      for (int j = i + 1; j < segCount; j++) {
        if (segments[j].x0 < segments[i].x0) {
          Segment tmp = segments[i];
          segments[i] = segments[j];
          segments[j] = tmp;
        }
      }
    }
 
    // Merge overlapping segments.
    float mergedStart = segments[0].x0;
    float mergedEnd = segments[0].x1;
    for (int i = 1; i < segCount; i++) {
      if (segments[i].x0 <= mergedEnd + 1) {  // overlapping or contiguous
        if (segments[i].x1 > mergedEnd)
          mergedEnd = segments[i].x1;
      } else {
        // Draw the current merged segment.
        int ix0 = (int)ceil(mergedStart);
        int ix1 = (int)floor(mergedEnd);
        if (ix1 >= ix0)
          canvas.drawFastHLine(ix0, y, ix1 - ix0 + 1, color);
        mergedStart = segments[i].x0;
        mergedEnd = segments[i].x1;
      }
    }
    // Draw the final merged segment.
    int ix0 = (int)ceil(mergedStart);
    int ix1 = (int)floor(mergedEnd);
    if (ix1 >= ix0)
      canvas.drawFastHLine(ix0, y, ix1 - ix0 + 1, color);
  }
}

void setup() {
  tft.init(320, 480, 0, 0, ST7796S_BGR);
  tft.setSPISpeed(40000000);
  tft.invertDisplay(true);
  tft.setRotation(rotate);
}

void loop() {
  canvas.fillScreen(blackish);

  const float baseThickness = 24.0;
  int segmentIndex = 0;
  float prev_px = -1, prev_py = -1;
  float prev_z = 0;
  int prev_px_i = 0, prev_py_i = 0;

  for (int i = 0; i <= NUM_SEGMENTS; ++i) {
    float t = (360.0 / NUM_SEGMENTS) * i;
    float rad = t * PI_F / 180.0;

    float x = SCALE * (cos(rad) - 0.5 * cos(5 * rad));
    float y = SCALE * (sin(rad) + 0.5 * sin(5 * rad));
    float z = Z_SCALE * cos(3 * rad);

    Vec3 point = { x, y, z };
    Vec3 rotated = rotateY(point, rotation);

    int px, py;
    project(rotated, px, py);

    if (i > 0 && px >= 0 && px < SCREEN_WIDTH && py >= 0 && py < SCREEN_HEIGHT) {
      float avgZ = (rotated.z + prev_z) / 2.0;
      float perspectiveScale = 1.0f / (Z_OFFSET - avgZ);
      int thickness = baseThickness * perspectiveScale;
      if (thickness < 1) thickness = 1;
      if (thickness > 20) thickness = 20;

      segments[segmentIndex++] = {
        prev_px_i, prev_py_i, px, py, avgZ, thickness
      };
    }

    prev_px_i = px;
    prev_py_i = py;
    prev_z = rotated.z;
  }

  for (int i = 0; i < segmentIndex - 1; ++i) {
    for (int j = i + 1; j < segmentIndex; ++j) {
      if (segments[j].avgZ < segments[i].avgZ) {
        Segment3D tmp = segments[i];
        segments[i] = segments[j];
        segments[j] = tmp;
      }
    }
  }

  for (int i = 0; i < segmentIndex; ++i) {
    float depthFactor = (Z_OFFSET - segments[i].avgZ - 1.5f);
    depthFactor = constrain(depthFactor, 0.0f, 1.0f);
    uint8_t brightness = 255 - (uint8_t)(depthFactor * 192);
    uint16_t color = rgb888_to_565(0, brightness, brightness);

    DrawThickLineRoundedHLines(
      canvas,
      segments[i].x1, segments[i].y1,
      segments[i].x2, segments[i].y2,
      segments[i].thickness,
      color
    );
  }

  tft.drawRGBBitmap(0, 0, canvas.getBuffer(), SCREEN_WIDTH, SCREEN_HEIGHT);
  rotation += 2.0;
  if (rotation >= 360.0) rotation -= 360.0;
}