以下是基于您提供的ultraviolet数学库内容的完整示例demo，包含物理模拟和几何运算的综合应用：

use ultraviolet as uv;

// 物理粒子系统模拟示例
fn particle_simulation() {
    const PARTICLE_COUNT: usize = 128;
    
    // 初始化SIMD向量(每个Vec3x8处理8个粒子)
    let mut positions = Vec::with_capacity(PARTICLE_COUNT / 8 + 1);
    let mut velocities = Vec::with_capacity(PARTICLE_COUNT / 8 + 1);
    let mut accelerations = Vec::with_capacity(PARTICLE_COUNT / 8 + 1);

    // 随机初始化粒子状态
    for i in 0..(PARTICLE_COUNT / 8 + 1) {
        let offset = i as f32 * 0.5;
        positions.push(uv::Vec3x8::new(
            uv::f32x8::splat(offset), 
            uv::f32x8::splat(0.0), 
            uv::f32x8::splat(0.0)
        ));
        
        velocities.push(uv::Vec3x8::new(
            uv::f32x8::splat(0.0),
            uv::f32x8::splat(0.1 + i as f32 * 0.01),
            uv::f32x8::splat(0.0)
        ));
        
        // 重力加速度
        accelerations.push(uv::Vec3x8::new(
            uv::f32x8::splat(0.0),
            uv::f32x8::splat(-9.8),
            uv::f32x8::splat(0.0)
        ));
    }

    // 模拟100帧
    let dt = uv::f32x8::splat(0.016); // 60fps
    for _ in 0..100 {
        // 欧拉积分更新位置和速度
        integrate_x8(&mut positions, &mut velocities, &accelerations, dt);
        
        // 边界检测和反弹
        for pos in &mut positions {
            let y_pos = pos.y;
            let below_ground = y_pos.cmp_lt(uv::f32x8::splat(0.0));
            pos.y = below_ground.blend(-y_pos, y_pos);
        }
    }
}

// 几何变换示例
fn geometric_transformations() {
    // 创建旋转器(Rotor) - 替代四元数
    let angle = std::f32::consts::PI / 4.0; // 45度
    let rotor = uv::Rotor3::from_rotation_xz(angle);
    
    // 创建平移向量
    let translation = uv::Vec3::new(1.0, 0.0, 1.0);
    
    // 创建缩放因子
    let scale = 2.0;
    
    // 组合变换
    let transform = uv::Isometry3::new(translation, rotor) * scale;
    
    // 变换点
    let point = uv::Vec3::new(1.0, 0.0, 0.0);
    let transformed_point = transform.transform_point(point);
    
    println!("Transformed point: {:?}", transformed_point);
}

// 光线追踪示例
fn ray_tracing_demo() {
    // 创建8个球体
    let sphere_centers = uv::Vec3x8::new(
        uv::f32x8::new(0.0, 0.0, 5.0, 1.0, -1.0, 2.0, -2.0, 3.0),
        uv::f32x8::new(0.0, 1.0, 4.0, -1.0, 2.0, -2.0, 3.0, 0.0),
        uv::f32x8::new(5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0)
    );
    
    let sphere_radii_sq = uv::f32x8::new(
        1.0, 1.5, 2.0, 0.5, 1.0, 1.5, 2.0, 0.5
    );
    
    // 创建光线
    let ray_origin = uv::Vec3x8::broadcast(uv::Vec3::new(0.0, 0.0, 0.0));
    let ray_direction = uv::Vec3x8::broadcast(uv::Vec3::new(0.0, 0.0, 1.0));
    
    // 批量检测相交
    let intersections = ray_sphere_intersect_x8(
        sphere_centers,
        sphere_radii_sq,
        ray_origin,
        ray_direction
    );
    
    // 提取结果
    let mut results = [0.0; 8];
    intersections.store(&mut results);
    println!("Intersection distances: {:?}", results);
}

fn main() {
    println!("Running particle simulation demo...");
    particle_simulation();
    
    println!("\nRunning geometric transformations demo...");
    geometric_transformations();
    
    println!("\nRunning ray tracing demo...");
    ray_tracing_demo();
}

// 以下是之前提供的工具函数
fn integrate_x8(
    pos: &mut [uv::Vec3x8],
    vel: &mut [uv::Vec3x8],
    acc: &[uv::Vec3x8],
    dt: uv::f32x8,
) {
    for ((position, velocity), acceleration) in pos.iter_mut().zip(vel).zip(acc) {
        *velocity = *velocity + *acceleration * dt;
        *position = *position + *velocity * dt;
    }
}

fn ray_sphere_intersect_x8(
    sphere_o: uv::Vec3x8,
    sphere_r_sq: uv::f32x8,
    ray_o: uv::Vec3x8,
    ray_d: uv::Vec3x8,
) -> uv::f32x8 {
    let oc = ray_o - sphere_o;
    let b = oc.dot(ray_d);
    let c = oc.mag_sq() - sphere_r_sq;
    let descrim = b * b - c;

    let desc_pos = descrim.cmp_gt(uv::f32x8::splat(0.0));
    let desc_sqrt = descrim.sqrt();

    let t1 = -b - desc_sqrt;
    let t1_valid = t1.cmp_gt(uv::f32x8::splat(0.0)) & desc_pos;

    let t2 = -b + desc_sqrt;
    let t2_valid = t2.cmp_gt(uv::f32x8::splat(0.0)) & desc_pos;

    let t = t2_valid.blend(t2, uv::f32x8::splat(std::f32::MAX));
    let t = t1_valid.blend(t1, t);

    t
}

这个完整示例演示了：

1. 粒子系统物理模拟（使用SIMD加速）
2. 几何变换（使用Rotor进行旋转）
3. 批量光线与球体相交检测

要运行此示例，确保在Cargo.toml中添加了ultraviolet依赖：

[dependencies]
ultraviolet = "0.10.0"