diff --git a/src/utils/cone.ts b/src/utils/cone.ts
index c0250183..0bbf4693 100644
--- a/src/utils/cone.ts
+++ b/src/utils/cone.ts
@@ -1,5 +1,8 @@
 import { calculateEquation2 } from "./equation-calculater"
-import { isZero } from "./math"
+import { equals, isZero } from "./math"
+import { v3 } from "./matrix"
+import { getPerpendicularPoint3D } from "./perpendicular"
+import { getPlaneByPointAndNormal, getPointAndDirectionLineAndPlaneIntersectionPoint, getPointAndTwoDirectionsPlane, rotateDirectionByRadianOnPlane } from "./plane"
 import { getPointByLengthAndDirection3D } from "./position"
 import { Vec3 } from "./types"
 import { number, Validator } from "./validators"
@@ -58,6 +61,21 @@ export function getLineAndConeIntersectionPoints([[x1, y1, z1], [x2, y2, z2]]: [
   ])
 }
 
+export function pointIsOnCone(p: Vec3, cone: Cone) {
+  const point = getPerpendicularPoint3D(p, cone.base, cone.direction)
+  const height = v3.length(v3.substract(point, cone.base))
+  const radius = v3.length(v3.substract(point, p))
+  return equals(radius, cone.radiusHeightRate * height)
+}
+
+export function getPerpendicularPointsToCone(p: Vec3, cone: Cone): Vec3[] | undefined {
+  const plane = getPointAndTwoDirectionsPlane(cone.base, cone.direction, v3.substract(p, cone.base))
+  if (!plane) return
+  const radian = Math.atan(cone.radiusHeightRate)
+  const directions = rotateDirectionByRadianOnPlane(cone.direction, radian, plane)
+  return directions.map(d => getPerpendicularPoint3D(p, cone.base, d))
+}
+
 export function getParallelConesByDistance(cone: Cone, distance: number): [Cone, Cone] {
   if (isZero(distance)) {
     return [cone, cone]
@@ -68,3 +86,10 @@ export function getParallelConesByDistance(cone: Cone, distance: number): [Cone,
     { ...cone, base: getPointByLengthAndDirection3D(cone.base, -length, cone.direction) },
   ]
 }
+
+export function getConeNormalAtPoint(cone: Cone, p: Vec3): Vec3 | undefined {
+  const plane = getPlaneByPointAndNormal(p, v3.substract(p, cone.base))
+  const point = getPointAndDirectionLineAndPlaneIntersectionPoint(cone.base, cone.direction, plane)
+  if (!point) return
+  return v3.substract(p, point)
+}
diff --git a/src/utils/cylinder.ts b/src/utils/cylinder.ts
index 3cbecee7..ef31bdb5 100644
--- a/src/utils/cylinder.ts
+++ b/src/utils/cylinder.ts
@@ -1,5 +1,5 @@
 import { calculateEquation2 } from "./equation-calculater";
-import { isZero } from "./math";
+import { equals, isZero } from "./math";
 import { v3 } from "./matrix";
 import { getPerpendicularPoint3D } from "./perpendicular";
 import { getPointByLengthAndDirection3D } from "./position";
@@ -55,6 +55,11 @@ export function getLineAndCylinderIntersectionPoints([[x1, y1, z1], [x2, y2, z2]
   ])
 }
 
+export function pointIsOnCylinder(p: Vec3, cylinder: Cylinder) {
+  const point = getPerpendicularPoint3D(p, cylinder.base, cylinder.direction)
+  return equals(v3.length(v3.substract(p, point)), cylinder.radius)
+}
+
 export function getPerpendicularPointsToCylinder(p: Vec3, cylinder: Cylinder): [Vec3, Vec3] {
   const point = getPerpendicularPoint3D(p, cylinder.base, cylinder.direction)
   const direction = v3.substract(point, p)
@@ -73,3 +78,8 @@ export function getParallelCylindersByDistance(cylinder: Cylinder, distance: num
     { ...cylinder, radius: cylinder.radius + distance },
   ]
 }
+
+export function getCylinderNormalAtPoint(cylinder: Cylinder, p: Vec3): Vec3 {
+  const point = getPerpendicularPoint3D(p, cylinder.base, cylinder.direction)
+  return v3.substract(p, point)
+}
diff --git a/src/utils/matrix.ts b/src/utils/matrix.ts
index 3cfc0eec..1bef579d 100644
--- a/src/utils/matrix.ts
+++ b/src/utils/matrix.ts
@@ -262,7 +262,10 @@ export const v3 = {
     return [a[0] - b[0], a[1] - b[1], a[2] - b[2]]
   },
   length(a: Vec3): number {
-    return Math.sqrt(a[0] ** 2 + a[1] ** 2 + a[2] ** 2)
+    return Math.sqrt(v3.lengthSquare(a))
+  },
+  lengthSquare(a: Vec3): number {
+    return a[0] ** 2 + a[1] ** 2 + a[2] ** 2
   },
   multipleScalar(a: Vec3, b: number): Vec3 {
     return [a[0] + b, a[1] + b, a[2] + b]
diff --git a/src/utils/plane.ts b/src/utils/plane.ts
index 698be893..9a6cb40e 100644
--- a/src/utils/plane.ts
+++ b/src/utils/plane.ts
@@ -1,3 +1,4 @@
+import { calculateEquation2 } from "./equation-calculater";
 import { GeneralFormLine } from "./line";
 import { isZero } from "./math";
 import { v3 } from "./matrix";
@@ -49,34 +50,49 @@ export function pointIsOnPlane([x, y, z]: Vec3, { a, b, c, d }: GeneralFormPlane
   return isZero(a * x + b * y + c * z + d)
 }
 
-export function getThreePointPlane([x1, y1, z1]: Vec3, [x2, y2, z2]: Vec3, [x3, y3, z3]: Vec3): GeneralFormPlane | undefined {
+export function directionIsOnPlane([x, y, z]: Vec3, { a, b, c }: GeneralFormPlane) {
+  // a x0 + b y0 + c z0 + d = 0
+  // a (x0 + x) + b (y0 + y) + c (z0 + z) + d = 0
+  // a x + b y + c z = 0
+  return isZero(a * x + b * y + c * z)
+}
+
+export function getThreePointPlane(p1: Vec3, p2: Vec3, p3: Vec3): GeneralFormPlane | undefined {
+  return getPointAndTwoDirectionsPlane(p1, v3.substract(p2, p1), v3.substract(p3, p1))
+}
+
+export function getPointAndTwoDirectionsPlane([x1, y1, z1]: Vec3, [e1, f1, g1]: Vec3, [e2, f2, g2]: Vec3): GeneralFormPlane | undefined {
   // a x1 + b y1 + c z1 + d = 0
-  // a x2 + b y2 + c z2 + d = 0
-  // a x3 + b y3 + c z3 + d = 0
-  const e1 = x1 - x2, e2 = x1 - x3
-  const f1 = y1 - y2, f2 = y1 - y3
-  const g1 = z1 - z2, g2 = z1 - z3
+  // a(x1 + e1) + b(y1 + f1) + c(z1 + g1) + d = 0
+  // a(x1 + e2) + b(y1 + f2) + c(z1 + g2) + d = 0
+
   // a e1 + b f1 + c g1 = 0
   // a e2 + b f2 + c g2 = 0
 
   // a e1 g2 + b f1 g2 + c g1 g2 = 0
   // a e2 g1 + b f2 g1 + c g1 g2 = 0
+  // a(e2 g1 - e1 g2) - b(f1 g2 - f2 g1) = 0
+
+  // a e1 f2 + b f1 f2 + c f2 g1 = 0
+  // a e2 f1 + b f1 f2 + c f1 g2 = 0
+  // a(e1 f2 - e2 f1) - c(f1 g2 - f2 g1) = 0
 
-  // a (e1 g2 - e2 g1) + b (f1 g2 - f2 g1) = 0
   const a = f1 * g2 - f2 * g1
   const b = e2 * g1 - e1 * g2
-  let c: number
-  if (!isZero(g1)) {
-    c = -(a * e1 + b * f1) / g1
-  } else if (!isZero(g2)) {
-    c = -(a * e2 + b * f2) / g2
-  } else {
-    return { a: 0, b: 0, c: 1, d: -z1 }
-  }
+  const c = e1 * f2 - e2 * f1
   const d = -(a * x1 + b * y1 + c * z1)
   return { a, b, c, d }
 }
 
+export function getPlaneByPointAndNormal([x, y, z]: Vec3, [a, b, c]: Vec3): GeneralFormPlane {
+  return {
+    a,
+    b,
+    c,
+    d: -(a * x + b * y + c * z),
+  }
+}
+
 export function pointInTriangle(p: Vec3, a: Vec3, b: Vec3, c: Vec3): boolean {
   // p = a + u(c - a) + v(b - a)
   const v0 = v3.substract(c, a)
@@ -151,3 +167,38 @@ export function getLineAndTrianglesIntersectionPoint(line: [Vec3, Vec3], triangl
   }
   return points
 }
+
+export function rotateDirectionByRadianOnPlane(direction: Vec3, radian: number, { a, b, c }: GeneralFormPlane): Vec3[] {
+  const [d1, d2, d3] = direction
+  const e = Math.cos(radian)
+  const f = v3.lengthSquare(direction)
+  // a x + b y + c z = 0
+  // x^2 + y^2 + z^2 = f
+  // d1 x + d2 y + d3 z = f e
+
+  // a d3 x + b d3 y + c d3 z = 0
+  // c d1 x + c d2 y + c d3 z = c f e
+  // (c d1 - a d3) x + (c d2 - b d3) y = c f e
+  const g1 = c * d1 - a * d3, g2 = c * d2 - b * d3, g3 = c * f * e
+  // g1 x + g2 y - g3 = 0
+  // g2 y = g3 - g1 x
+
+  // c z = -(a x + b y)
+  // c c x^2 + c c y^2 + c c z^2 = c c f
+  // c c x^2 + c c y^2 + (a x + b y)^2 - c c f = 0
+  // (a a + c c) x x + 2 a b x y + (b b + c c) y y - c c f = 0
+  const h1 = a * a + c * c, h2 = b * b + c * c, h3 = c * c * f
+  // h1 x x + 2 a b x y + h2 y y - h3 = 0
+  // *g2 g2: g2 g2 h1 x x + 2 a b g2 g2 x y + g2 g2 h2 y y - g2 g2 h3 = 0
+  // g2 g2 h1 x x + 2 a b g2 x(g3 - g1 x) + h2(g3 - g1 x)^2 - g2 g2 h3 = 0
+  // (-2 a b g1 g2 + g2 g2 h1 + g1 g1 h2) x x + (2 a b g2 g3 + -2 g1 g3 h2) x + g3 g3 h2 + -g2 g2 h3 = 0
+  const xs = calculateEquation2(
+    -2 * a * b * g1 * g2 + g2 * g2 * h1 + g1 * g1 * h2,
+    2 * a * b * g2 * g3 + -2 * g1 * g3 * h2,
+    g3 * g3 * h2 + -1 * g2 * g2 * h3
+  )
+  return xs.map(x => {
+    const y = (g3 - g1 * x) / g2
+    return [x, y, -(a * x + b * y) / c]
+  })
+}
diff --git a/src/utils/sphere.ts b/src/utils/sphere.ts
index 5230df2b..e81f8c17 100644
--- a/src/utils/sphere.ts
+++ b/src/utils/sphere.ts
@@ -1,6 +1,7 @@
 import { calculateEquation2 } from "./equation-calculater"
-import { isZero } from "./math"
-import { Position3D, getPointByLengthAndDirection3D } from "./position"
+import { equals, isZero } from "./math"
+import { v3 } from "./matrix"
+import { Position3D, getPointByLengthAndDirection3D, getTwoPointsDistance3D, position3DToVec3, vec3ToPosition3D } from "./position"
 import { Vec3 } from "./types"
 import { and, number } from "./validators"
 
@@ -39,6 +40,10 @@ export function getLineAndSphereIntersectionPoints([[x1, y1, z1], [x2, y2, z2]]:
   ])
 }
 
+export function pointIsOnSphere(p: Vec3, sphere: Sphere) {
+  return equals(getTwoPointsDistance3D(vec3ToPosition3D(p), sphere), sphere.radius)
+}
+
 export function getPerpendicularPointsToSphere(p: Vec3, sphere: Sphere): [Vec3, Vec3] {
   const direction: Vec3 = [
     p[0] - sphere.x,
@@ -60,3 +65,7 @@ export function getParallelSpheresByDistance(sphere: Sphere, distance: number):
     { ...sphere, radius: sphere.radius + distance },
   ]
 }
+
+export function getSphereNormalAtPoint(sphere: Sphere, point: Vec3): Vec3 {
+  return v3.substract(point, position3DToVec3(sphere))
+}